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id="toc-Nomenclature-sublist" class="vector-toc-list"> <li id="toc-Equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Equations</span> </div> </a> <ul id="toc-Equations-sublist" class="vector-toc-list"> <li id="toc-Tangent_as_a_ratio" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Tangent_as_a_ratio"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.1</span> <span>Tangent as a ratio</span> </div> </a> <ul id="toc-Tangent_as_a_ratio-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Angle_from_a_tangent_gradient" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Angle_from_a_tangent_gradient"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.2</span> <span>Angle from a tangent gradient</span> </div> </a> <ul id="toc-Angle_from_a_tangent_gradient-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Example_slopes_comparing_the_notations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example_slopes_comparing_the_notations"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Example slopes comparing the notations</span> </div> </a> <ul id="toc-Example_slopes_comparing_the_notations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Roads" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Roads"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Roads</span> </div> </a> <ul id="toc-Roads-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Environmental_design" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Environmental_design"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Environmental design</span> </div> </a> <ul id="toc-Environmental_design-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Railways" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Railways"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Railways</span> </div> </a> <button aria-controls="toc-Railways-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Railways subsection</span> </button> <ul id="toc-Railways-sublist" class="vector-toc-list"> <li id="toc-Compensation_for_curvature" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Compensation_for_curvature"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Compensation for curvature</span> </div> </a> <ul id="toc-Compensation_for_curvature-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Continuous_brakes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Continuous_brakes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Continuous brakes</span> </div> </a> <ul id="toc-Continuous_brakes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" 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href="https://bg.wikipedia.org/wiki/%D0%9D%D0%B0%D0%BA%D0%BB%D0%BE%D0%BD" title="Наклон – Bulgarian" lang="bg" hreflang="bg" data-title="Наклон" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Pendent_(topografia)" title="Pendent (topografia) – Catalan" lang="ca" hreflang="ca" data-title="Pendent (topografia)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Stoup%C3%A1n%C3%AD" title="Stoupání – Czech" lang="cs" hreflang="cs" data-title="Stoupání" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Muteremuko" title="Muteremuko – Shona" lang="sn" hreflang="sn" data-title="Muteremuko" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Gradiente" title="Gradiente – German" lang="de" hreflang="de" data-title="Gradiente" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kalle" title="Kalle – Estonian" lang="et" hreflang="et" data-title="Kalle" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Pendiente_(geograf%C3%ADa)" title="Pendiente (geografía) – Spanish" lang="es" hreflang="es" data-title="Pendiente (geografía)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Malda_(topografia)" title="Malda (topografia) – Basque" lang="eu" hreflang="eu" data-title="Malda (topografia)" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D8%B1%D8%AC%D9%87_(%D8%B4%DB%8C%D8%A8)" title="درجه (شیب) – Persian" lang="fa" hreflang="fa" data-title="درجه (شیب)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Pente_(topographie)" title="Pente (topographie) – French" lang="fr" hreflang="fr" data-title="Pente (topographie)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B2%BD%EC%82%AC_(%EC%A7%80%EB%A6%AC)" title="경사 (지리) – Korean" lang="ko" hreflang="ko" data-title="경사 (지리)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Derajat_(kemiringan)" title="Derajat (kemiringan) – Indonesian" lang="id" hreflang="id" data-title="Derajat (kemiringan)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Pendenza_topografica" title="Pendenza topografica – Italian" lang="it" hreflang="it" data-title="Pendenza topografica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/%C5%A0laitas" title="Šlaitas – Lithuanian" lang="lt" hreflang="lt" data-title="Šlaitas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Helling_(geografie)" title="Helling (geografie) – Dutch" lang="nl" hreflang="nl" data-title="Helling (geografie)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%B8%A6%E6%96%AD%E5%8B%BE%E9%85%8D" title="縦断勾配 – Japanese" lang="ja" hreflang="ja" data-title="縦断勾配" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Stigning" title="Stigning – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Stigning" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nrm mw-list-item"><a href="https://nrm.wikipedia.org/wiki/C%C3%B4ti" title="Côti – Norman" lang="nrf" hreflang="nrf" data-title="Côti" data-language-autonym="Nouormand" data-language-local-name="Norman" class="interlanguage-link-target"><span>Nouormand</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Pochylenie_pod%C5%82u%C5%BCne" title="Pochylenie podłużne – Polish" lang="pl" hreflang="pl" data-title="Pochylenie podłużne" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Talude" title="Talude – Portuguese" lang="pt" hreflang="pt" data-title="Talude" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A3%D0%BA%D0%BB%D0%BE%D0%BD_(%D0%B3%D0%B5%D0%BE%D0%B4%D0%B5%D0%B7%D0%B8%D1%8F)" title="Уклон (геодезия) – Russian" lang="ru" hreflang="ru" data-title="Уклон (геодезия)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Pinninu_(nclinazzioni)" title="Pinninu (nclinazzioni) – Sicilian" lang="scn" hreflang="scn" data-title="Pinninu (nclinazzioni)" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Grade_(slope)" title="Grade (slope) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Grade (slope)" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Lutningsf%C3%B6rh%C3%A5llande" title="Lutningsförhållande – Swedish" lang="sv" hreflang="sv" data-title="Lutningsförhållande" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%BE%D1%85%D0%B8%D0%BB_(%D0%BF%D0%BE%D0%BB%D0%BE%D0%B6%D0%B5%D0%BD%D0%BD%D1%8F,_%D0%BF%D0%BE%D0%BA%D0%B0%D0%B6%D1%87%D0%B8%D0%BA_%D0%BA%D1%80%D1%83%D1%82%D0%B8%D0%B7%D0%BD%D0%B8)" title="Похил (положення, покажчик крутизни) – Ukrainian" lang="uk" hreflang="uk" data-title="Похил (положення, покажчик крутизни)" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%96%9C%E5%9D%A1" title="斜坡 – Cantonese" lang="yue" hreflang="yue" data-title="斜坡" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Atkaln%C4%97" title="Atkalnė – Samogitian" lang="sgs" hreflang="sgs" data-title="Atkalnė" data-language-autonym="Žemaitėška" data-language-local-name="Samogitian" class="interlanguage-link-target"><span>Žemaitėška</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%9D%A1%E5%BA%A6" title="坡度 – Chinese" lang="zh" hreflang="zh" data-title="坡度" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1139344#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> 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data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Angle to the horizontal plane</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about the grade of a topographic feature or constructed element. For other uses, see <a href="/wiki/Slope_(disambiguation)" class="mw-disambig" title="Slope (disambiguation)">Slope (disambiguation)</a>.</div> <p class="mw-empty-elt"> </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Grade_dimension.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Grade_dimension.svg/250px-Grade_dimension.svg.png" decoding="async" width="250" height="102" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Grade_dimension.svg/375px-Grade_dimension.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Grade_dimension.svg/500px-Grade_dimension.svg.png 2x" data-file-width="400" data-file-height="163" /></a><figcaption>d = run <br />Δh = rise <br />l = slope length <br />α = angle of inclination</figcaption></figure> <p>The <b>grade</b> (US) or <b>gradient</b> (UK) (also called <b>stepth</b>, <b>slope</b>, <b>incline</b>, <b>mainfall</b>, <b>pitch</b> or <b>rise</b>) of a physical feature, landform or constructed line is either the <a href="/wiki/Elevation_angle" class="mw-redirect" title="Elevation angle">elevation angle</a> of that surface to the <a href="/wiki/Horizontal_plane" class="mw-redirect" title="Horizontal plane">horizontal</a> or its tangent. It is a special case of the <a href="/wiki/Slope" title="Slope">slope</a>, where zero indicates <a href="/wiki/Horizontal_plane" class="mw-redirect" title="Horizontal plane">horizontality</a>. A larger number indicates higher or steeper degree of "tilt". Often slope is calculated as a ratio of "rise" to "run", or as a fraction ("rise over run") in which <i>run</i> is the horizontal distance (not the distance along the slope) and <i>rise</i> is the vertical distance. </p><p>Slopes of existing physical features such as <a href="/wiki/Canyon" title="Canyon">canyons</a> and hillsides, <a href="/wiki/Bank_(geography)" title="Bank (geography)">stream and river banks</a>, and <a href="/wiki/Stream_bed" title="Stream bed">beds</a> are often described as grades, but typically the word "grade" is used for human-made surfaces such as roads, <a href="/wiki/Landscape" title="Landscape">landscape</a> grading, <a href="/wiki/Roof_pitch" title="Roof pitch">roof pitches</a>, <a href="/wiki/Rail_tracks" class="mw-redirect" title="Rail tracks">railroads</a>, <a href="/wiki/Aqueduct_(watercourse)" class="mw-redirect" title="Aqueduct (watercourse)">aqueducts</a>, and pedestrian or bicycle routes. The grade may refer to the <a href="/wiki/Geometric_terms_of_location" title="Geometric terms of location">longitudinal</a> slope or the <a href="/wiki/Geometric_terms_of_location" title="Geometric terms of location">perpendicular</a> <a href="/wiki/Cross_slope" title="Cross slope">cross slope</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Nomenclature">Nomenclature</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grade_(slope)&action=edit&section=1" title="Edit section: Nomenclature"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Slope_quadrant.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Slope_quadrant.svg/330px-Slope_quadrant.svg.png" decoding="async" width="330" height="330" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Slope_quadrant.svg/495px-Slope_quadrant.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Slope_quadrant.svg/660px-Slope_quadrant.svg.png 2x" data-file-width="540" data-file-height="540" /></a><figcaption>Illustration of grades (percentages), angles in degrees, and ratio.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:50_per_mille.png" class="mw-file-description"><img alt="A railroad distance and gradient sign in Gdańsk, Poland. The 50‰ grade is equivalent to 5%." src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/50_per_mille.png/220px-50_per_mille.png" decoding="async" width="220" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/50_per_mille.png/330px-50_per_mille.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/50_per_mille.png/440px-50_per_mille.png 2x" data-file-width="1866" data-file-height="1420" /></a><figcaption>A tram and gradient sign in <a href="/wiki/Gda%C5%84sk,_Poland" class="mw-redirect" title="Gdańsk, Poland">Gdańsk, Poland</a>. The 50‰ grade is equivalent to 50<span class="nowrap"> </span>m/km or 5%.</figcaption></figure> <p>There are several ways to express slope: </p> <ol><li>as an <i>angle</i> of inclination to the horizontal. (This is the angle <span class="texhtml mvar" style="font-style:italic;">α</span> opposite the "rise" side of a triangle with a right angle between vertical rise and horizontal run.)</li> <li>as a <i><a href="/wiki/Percent_sign" title="Percent sign">percentage</a></i>, the formula for which is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 100\times {\frac {\text{rise}}{\text{run}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>100</mn> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>rise</mtext> <mtext>run</mtext> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 100\times {\frac {\text{rise}}{\text{run}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c97f57441d5cd42da361570e6a1afddb80b6a860" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.671ex; height:5.176ex;" alt="{\displaystyle 100\times {\frac {\text{rise}}{\text{run}}}}"></span> which is equivalent to the tangent of the angle of inclination times 100. In Europe and the U.S. percentage "grade" is the most commonly used figure for describing slopes.</li> <li>as a <i><a href="/wiki/Per_mille" title="Per mille">per mille</a></i> figure (denoted with the symbol <span class="nounderlines" style="border: 1px solid var(--border-color-muted,#ddd); color: var(--color-base,#202122); background-color: var( --background-color-neutral-subtle, #fdfdfd); padding: 1px 1px;">‰</span>), the formula for which is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1000\times {\frac {\text{rise}}{\text{run}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1000</mn> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>rise</mtext> <mtext>run</mtext> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1000\times {\frac {\text{rise}}{\text{run}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01dae2ef6420d3b310babfcc0f16488f2c4c1f27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.833ex; height:5.176ex;" alt="{\displaystyle 1000\times {\frac {\text{rise}}{\text{run}}}}"></span> which could also be expressed as the tangent of the angle of inclination times 1000. This is commonly used in Europe to denote the incline of a railway. It is sometimes written as mm/m or m/km instead of the ‰ symbol.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li> <li>as a <i>ratio</i> of one part rise to so many parts run. For example, a slope that has a rise of 5 feet for every 1000 feet of run would have a slope ratio of 1 in 200. (The word "in" is normally used rather than the mathematical ratio notation of "1:200".) This is generally the method used to describe railway grades in Australia and the UK. It is used for roads in Hong Kong, and was used for roads in the UK until the 1970s.</li> <li>as a <i>ratio</i> of many parts run to one part rise, which is the inverse of the previous expression (depending on the country and the industry standards). For example, "slopes are expressed as ratios such as 4:1. This means that for every 4 units (feet or metres) of horizontal distance there is a 1 unit (foot or metre) vertical change either up or down."<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li></ol> <p>Any of these may be used. Grade is usually expressed as a percentage - converted to the angle <span class="texhtml mvar" style="font-style:italic;">α</span> by taking the inverse tangent of the standard mathematical slope, which is rise / run or the grade / 100. If one looks at red numbers on the chart specifying grade, one can see the quirkiness of using the grade to specify slope; the numbers go from 0 for flat, to 100% at 45 degrees, to infinity as it approaches vertical. </p><p>Slope may still be expressed when the horizontal run is not known: the rise can be divided by the <a href="/wiki/Hypotenuse" title="Hypotenuse">hypotenuse</a> (the slope length). This is not the usual way to specify slope; this nonstandard expression follows the <a href="/wiki/Sine" class="mw-redirect" title="Sine">sine</a> function rather than the tangent function, so it calls a 45 degree slope a 71 percent grade instead of a 100 percent. But in practice the usual way to calculate slope is to measure the distance along the slope and the vertical rise, and calculate the horizontal run from that, in order to calculate the grade (100% × rise/run) or standard slope (rise/run). When the angle of inclination is small, using the slope length rather than the horizontal displacement (i.e., using the sine of the angle rather than the tangent) makes only an insignificant difference and can then be used as an approximation. Railway gradients are often expressed in terms of the rise in relation to the distance along the track as a practical measure. In cases where the difference between sin and tan is significant, the tangent is used. In either case, the following identity holds for all inclinations up to 90 degrees: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan {\alpha }={\frac {\sin {\alpha }}{\sqrt {1-\sin ^{2}{\alpha }}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan {\alpha }={\frac {\sin {\alpha }}{\sqrt {1-\sin ^{2}{\alpha }}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0395c661ed66f985d09d31f12917d1b68625bcac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.28ex; height:6.509ex;" alt="{\displaystyle \tan {\alpha }={\frac {\sin {\alpha }}{\sqrt {1-\sin ^{2}{\alpha }}}}}"></span>. Or more simply, one can calculate the horizontal run by using the Pythagorean theorem, after which it is trivial to calculate the (standard math) slope or the grade (percentage). </p><p>In Europe, road gradients are signed as a percentage.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Equations">Equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grade_(slope)&action=edit&section=2" title="Edit section: Equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Grades are related using the following equations with symbols from the figure at top. </p> <div class="mw-heading mw-heading4"><h4 id="Tangent_as_a_ratio">Tangent as a ratio</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grade_(slope)&action=edit&section=3" title="Edit section: Tangent as a ratio"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan {\alpha }={\frac {\Delta h}{d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>h</mi> </mrow> <mi>d</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan {\alpha }={\frac {\Delta h}{d}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a39121dcb37c2330e02815a81acd771d063c2b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.444ex; height:5.509ex;" alt="{\displaystyle \tan {\alpha }={\frac {\Delta h}{d}}}"></span></dd></dl> <p>The slope expressed as a percentage can similarly be determined from the tangent of the angle: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \%\,{\text{slope}}=100\tan {\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">%</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>slope</mtext> </mrow> <mo>=</mo> <mn>100</mn> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \%\,{\text{slope}}=100\tan {\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/777e773560e48a09eff5f5aa1e9bc9a66ac15d61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.581ex; height:2.676ex;" alt="{\displaystyle \%\,{\text{slope}}=100\tan {\alpha }}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Angle_from_a_tangent_gradient">Angle from a tangent gradient</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grade_(slope)&action=edit&section=4" title="Edit section: Angle from a tangent gradient"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\arctan {\frac {\Delta h}{d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>h</mi> </mrow> <mi>d</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =\arctan {\frac {\Delta h}{d}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2f9edc475e68a20c2c47f2020102b05606f5cda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.55ex; height:5.509ex;" alt="{\displaystyle \alpha =\arctan {\frac {\Delta h}{d}}}"></span></dd></dl> <p>If the tangent is expressed as a percentage, the angle can be determined as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\arctan {\frac {\%\,{\text{slope}}}{100}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">%</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>slope</mtext> </mrow> </mrow> <mn>100</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =\arctan {\frac {\%\,{\text{slope}}}{100}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cdd59bd2f36963680dce428ffbc0f93b6c9ece0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.649ex; height:5.509ex;" alt="{\displaystyle \alpha =\arctan {\frac {\%\,{\text{slope}}}{100}}}"></span></dd></dl> <p>If the angle is expressed as a ratio <i>(1 in n)</i> then: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\arctan {\frac {1}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =\arctan {\frac {1}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/316892bc968d0feb1e1b7a2cafffbb8179aad1bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.67ex; height:5.176ex;" alt="{\displaystyle \alpha =\arctan {\frac {1}{n}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Example_slopes_comparing_the_notations">Example slopes comparing the notations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grade_(slope)&action=edit&section=5" title="Edit section: Example slopes comparing the notations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For degrees, percentage (%) and per-mille (‰) notations, larger numbers are steeper slopes. For ratios, larger numbers <i>n</i> of 1 in <i>n</i> are shallower, easier slopes. </p><p>The examples show round numbers in one or more of the notations and some documented and reasonably well known instances. </p> <table class="wikitable"> <caption>Examples of slopes in the various notations </caption> <tbody><tr> <th>Degrees</th> <th>Percentage (%)</th> <th>Permillage (‰)</th> <th>Ratio</th> <th>Remarks </th></tr> <tr> <td>60°</td> <td>173%</td> <td>1732‰</td> <td>1 in 0.58</td> <td> </td></tr> <tr> <td>47.7°</td> <td>110%</td> <td>1100‰</td> <td>1 in 0.91</td> <td><a href="/wiki/Stoosbahn" title="Stoosbahn">Stoosbahn</a> (<a href="/wiki/Funicular_railway" class="mw-redirect" title="Funicular railway">funicular railway</a>) </td></tr> <tr> <td>45°</td> <td>100%</td> <td>1000‰</td> <td>1 in 1</td> <td> </td></tr> <tr> <td>35°</td> <td>70%</td> <td>700‰</td> <td>1 in 1.428</td> <td> </td></tr> <tr> <td>30.1°</td> <td>58%</td> <td>580‰</td> <td>1 in 1.724</td> <td><a href="/wiki/Lynton_and_Lynmouth_Cliff_Railway" title="Lynton and Lynmouth Cliff Railway">Lynton and Lynmouth Cliff Railway</a> (<a href="/wiki/Funicular_railway" class="mw-redirect" title="Funicular railway">funicular railway</a>) </td></tr> <tr> <td>30°</td> <td>58%</td> <td>577‰</td> <td>1 in 1.73</td> <td> </td></tr> <tr> <td>25.5°</td> <td>47%</td> <td>476‰</td> <td>1 in 2.1</td> <td><a href="/wiki/Pilatus_Railway" title="Pilatus Railway">Pilatus Railway</a> (steepest <a href="/wiki/Rack_railway" title="Rack railway">rack railway</a>) </td></tr> <tr> <td>20.3°</td> <td>37%</td> <td>370‰</td> <td>1 in 2.70</td> <td><a href="/wiki/Mount_Washington_Cog_Railway" title="Mount Washington Cog Railway">Mount Washington Cog Railway</a> (maximum grade) </td></tr> <tr> <td>20°</td> <td>36%</td> <td>363‰</td> <td>1 in 2.75</td> <td> </td></tr> <tr> <td>18.4°</td> <td>33%</td> <td>333‰</td> <td>1 in 3</td> <td> </td></tr> <tr> <td>16.9°</td> <td>30%</td> <td>300‰</td> <td>1 in 3.3</td> <td>Extremely steep road <figure class="mw-halign-left" typeof="mw:File/Frameless"><a href="/wiki/File:Pente30%25.jpg" class="mw-file-description" title="30% warning (France)"><img alt="30% warning (France)" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Pente30%25.jpg/22px-Pente30%25.jpg" decoding="async" width="22" height="30" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Pente30%25.jpg/34px-Pente30%25.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Pente30%25.jpg/45px-Pente30%25.jpg 2x" data-file-width="375" data-file-height="500" /></a><figcaption>30% warning (France)</figcaption></figure> </td></tr> <tr> <td>14.0°</td> <td>25%</td> <td>250‰</td> <td>1 in 4</td> <td>Very steep road. <a href="/wiki/Mount_Washington_Cog_Railway" title="Mount Washington Cog Railway">Mount Washington Cog Railway</a> (average grade) <figure class="mw-halign-left" typeof="mw:File/Frameless"><a href="/wiki/File:Devil%27s_Staircase_Wales.jpg" class="mw-file-description" title="25% warning (Wales)"><img alt="25% warning (Wales)" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Devil%27s_Staircase_Wales.jpg/30px-Devil%27s_Staircase_Wales.jpg" decoding="async" width="30" height="23" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Devil%27s_Staircase_Wales.jpg/45px-Devil%27s_Staircase_Wales.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Devil%27s_Staircase_Wales.jpg/60px-Devil%27s_Staircase_Wales.jpg 2x" data-file-width="2272" data-file-height="1704" /></a><figcaption>25% warning (Wales)</figcaption></figure> </td></tr> <tr> <td>11.3°</td> <td>20%</td> <td>200‰</td> <td>1 in 5</td> <td>Steep road <figure class="mw-halign-left" typeof="mw:File/Frameless"><a href="/wiki/File:Steep_Hill_sign_on_Henside_Road_-_geograph.org.uk_-_680341.jpg" class="mw-file-description" title="20% warning (North Yorkshire)"><img alt="20% warning (North Yorkshire)" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Steep_Hill_sign_on_Henside_Road_-_geograph.org.uk_-_680341.jpg/30px-Steep_Hill_sign_on_Henside_Road_-_geograph.org.uk_-_680341.jpg" decoding="async" width="30" height="23" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Steep_Hill_sign_on_Henside_Road_-_geograph.org.uk_-_680341.jpg/45px-Steep_Hill_sign_on_Henside_Road_-_geograph.org.uk_-_680341.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/82/Steep_Hill_sign_on_Henside_Road_-_geograph.org.uk_-_680341.jpg/60px-Steep_Hill_sign_on_Henside_Road_-_geograph.org.uk_-_680341.jpg 2x" data-file-width="640" data-file-height="480" /></a><figcaption>20% warning (North Yorkshire)</figcaption></figure> </td></tr> <tr> <td>8.13°</td> <td>14.2%</td> <td>142‰</td> <td>1 in 7</td> <td> </td></tr> <tr> <td>7.12°</td> <td>12.5%</td> <td>125‰</td> <td>1 in 8</td> <td><a href="/wiki/Cable_railway" title="Cable railway">Cable incline</a> on the <a href="/wiki/Cromford_and_High_Peak_Railway#Construction" title="Cromford and High Peak Railway">Cromford and High Peak Railway</a> </td></tr> <tr> <td>5.71°</td> <td>10%</td> <td>100‰</td> <td>1 in 10</td> <td>Steep road <figure class="mw-halign-left" typeof="mw:File/Frameless"><a href="/wiki/File:Nederlands_verkeersbord_J6.svg" class="mw-file-description" title="10% warning (Netherlands)"><img alt="10% warning (Netherlands)" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Nederlands_verkeersbord_J6.svg/30px-Nederlands_verkeersbord_J6.svg.png" decoding="async" width="30" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Nederlands_verkeersbord_J6.svg/45px-Nederlands_verkeersbord_J6.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/13/Nederlands_verkeersbord_J6.svg/60px-Nederlands_verkeersbord_J6.svg.png 2x" data-file-width="350" data-file-height="300" /></a><figcaption>10% warning (Netherlands)</figcaption></figure> </td></tr> <tr> <td>4.0°</td> <td>7%</td> <td>70‰</td> <td>1 in 14.3</td> <td> </td></tr> <tr> <td>3.37°</td> <td>5.9%</td> <td>59‰</td> <td>1 in 17</td> <td>Swannington <a href="/wiki/Cable_railway" title="Cable railway">incline</a> on the <a href="/wiki/Leicester_and_Swannington_Railway#Construction" title="Leicester and Swannington Railway">Leicester and Swannington Railway</a> </td></tr> <tr> <td>2.86°</td> <td>5%</td> <td>50‰</td> <td>1 in 20</td> <td><a href="/wiki/Matheran_Hill_Railway" title="Matheran Hill Railway">Matheran Hill Railway</a>. The incline from the <a href="/wiki/Crawlerway" title="Crawlerway">Crawlerway</a> at the <a href="/wiki/Kennedy_Space_Center" title="Kennedy Space Center">Kennedy Space Center</a> to the launch pads.<sup id="cite_ref-NASAsci_5-0" class="reference"><a href="#cite_note-NASAsci-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-facts-transp_6-0" class="reference"><a href="#cite_note-facts-transp-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td>2.29°</td> <td>4%</td> <td>40‰</td> <td>1 in 25</td> <td><a href="/wiki/Cologne%E2%80%93Frankfurt_high-speed_rail_line" title="Cologne–Frankfurt high-speed rail line">Cologne–Frankfurt high-speed rail line</a> </td></tr> <tr> <td>2.0°</td> <td>3.5%</td> <td>35‰</td> <td>1 in 28.57</td> <td><a href="/wiki/LGV_Sud-Est" title="LGV Sud-Est">LGV Sud-Est</a>, <a href="/wiki/LGV_Est" title="LGV Est">LGV Est</a>, <a href="/wiki/LGV_M%C3%A9diterran%C3%A9e" title="LGV Méditerranée">LGV Méditerranée</a> </td></tr> <tr> <td>1.97°</td> <td>3.4%</td> <td>34‰</td> <td>1 in 29</td> <td>Bagworth <a href="/wiki/Cable_railway" title="Cable railway">incline</a> on the <a href="/wiki/Leicester_and_Swannington_Railway#Construction" title="Leicester and Swannington Railway">Leicester and Swannington Railway</a> </td></tr> <tr> <td>1.89°</td> <td>3.3%</td> <td>33‰</td> <td>1 in 30.3</td> <td>Rampe de Capvern on the <a href="/wiki/Toulouse%E2%80%93Bayonne_railway" title="Toulouse–Bayonne railway">Toulouse–Bayonne railway</a><span class="noprint" style="font-size:85%; font-style: normal;"> [<a href="https://fr.wikipedia.org/wiki/Ligne_de_Toulouse_%C3%A0_Bayonne#Rampe_de_Capvern" class="extiw" title="fr:Ligne de Toulouse à Bayonne">fr</a>]</span> </td></tr> <tr> <td>1.52°</td> <td>2.65%</td> <td>26.5‰</td> <td>1 in 37.7</td> <td><a href="/wiki/Lickey_Incline" title="Lickey Incline">Lickey Incline</a> </td></tr> <tr> <td>1.43°</td> <td>2.5%</td> <td>25‰</td> <td>1 in 40</td> <td><a href="/wiki/LGV_Atlantique" title="LGV Atlantique">LGV Atlantique</a>, <a href="/wiki/LGV_Nord" title="LGV Nord">LGV Nord</a>. The <a href="/wiki/Schiefe_Ebene" title="Schiefe Ebene">Schiefe Ebene</a>. </td></tr> <tr> <td>1.146°</td> <td>2%</td> <td>20‰</td> <td>1 in 50</td> <td>Railway near <a href="/wiki/J%C3%ADlov%C3%A9_u_Prahy" title="Jílové u Prahy">Jílové u Prahy</a>. <a href="/wiki/Devonshire_Tunnel" title="Devonshire Tunnel">Devonshire Tunnel</a> <figure class="mw-halign-left" typeof="mw:File/Frameless"><a href="/wiki/File:Sklon%C3%ADk-kles%C3%A1n%C3%AD.jpg" class="mw-file-description" title="20‰ Czech Republic"><img alt="20‰ Czech Republic" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Sklon%C3%ADk-kles%C3%A1n%C3%AD.jpg/30px-Sklon%C3%ADk-kles%C3%A1n%C3%AD.jpg" decoding="async" width="30" height="23" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Sklon%C3%ADk-kles%C3%A1n%C3%AD.jpg/45px-Sklon%C3%ADk-kles%C3%A1n%C3%AD.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Sklon%C3%ADk-kles%C3%A1n%C3%AD.jpg/60px-Sklon%C3%ADk-kles%C3%A1n%C3%AD.jpg 2x" data-file-width="2048" data-file-height="1536" /></a><figcaption>20‰ Czech Republic</figcaption></figure> </td></tr> <tr> <td>0.819°</td> <td>1.43%</td> <td>14.3‰</td> <td>1 in 70</td> <td><a href="/wiki/Waverley_Route#Topography_and_construction" title="Waverley Route">Waverley Route</a> </td></tr> <tr> <td>0.716°</td> <td>1.25%</td> <td>12.5‰</td> <td>1 in 80</td> <td>Ruling grade of a secondary main line. <a href="/wiki/Wellington_Bank,_Somerset" title="Wellington Bank, Somerset">Wellington Bank, Somerset</a> </td></tr> <tr> <td>0.637°</td> <td>1.11%</td> <td>11.11‰</td> <td>1 in 90</td> <td><a href="/wiki/Dove_Holes_Tunnel" title="Dove Holes Tunnel">Dove Holes Tunnel</a> </td></tr> <tr> <td>0.573°</td> <td>1%</td> <td>10‰</td> <td>1 in 100</td> <td><a href="/wiki/Settle%E2%80%93Carlisle_line#Construction" title="Settle–Carlisle line">The long drag on the Settle & Carlisle line</a> </td></tr> <tr> <td>0.458°</td> <td>0.8%</td> <td>8‰</td> <td>1 in 125</td> <td>Rampe de Guillerval </td></tr> <tr> <td>0.2865°</td> <td>0.5%</td> <td>5‰</td> <td>1 in 200</td> <td><a href="/wiki/Paris%E2%80%93Bordeaux_railway" title="Paris–Bordeaux railway">Paris–Bordeaux railway</a><span class="noprint" style="font-size:85%; font-style: normal;"> [<a href="https://fr.wikipedia.org/wiki/Ligne_de_Paris-Austerlitz_%C3%A0_Bordeaux-Saint-Jean#Tracé_et_profil_en_long" class="extiw" title="fr:Ligne de Paris-Austerlitz à Bordeaux-Saint-Jean">fr</a>]</span>, except for the rampe de <a href="/wiki/Guillerval_station" title="Guillerval station">Guillerval</a> </td></tr> <tr> <td>0.1719°</td> <td>0.3%</td> <td>3‰</td> <td>1 in 333</td> <td> </td></tr> <tr> <td>0.1146°</td> <td>0.2%</td> <td>2‰</td> <td>1 in 500</td> <td> </td></tr> <tr> <td>0.0868°</td> <td>0.1515%</td> <td>1.515‰</td> <td>1 in 660</td> <td><a href="/wiki/Great_Western_Main_Line#History" title="Great Western Main Line">Brunel's Billiard Table</a> - <a href="/wiki/Great_Western_Main_Line#Infrastructure" title="Great Western Main Line">Didcot to Swindon</a> </td></tr> <tr> <td>0.0434°</td> <td>0.07575%</td> <td>0.7575‰</td> <td>1 in 1320</td> <td><a href="/wiki/Great_Western_Main_Line#History" title="Great Western Main Line">Brunel's Billiard Table</a> - <a href="/wiki/Great_Western_Main_Line#Infrastructure" title="Great Western Main Line">Paddington to Didcot</a> </td></tr> <tr> <td>0°</td> <td>0%</td> <td>0‰</td> <td>1 in ∞ (infinity)</td> <td>Flat </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Roads">Roads</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grade_(slope)&action=edit&section=6" title="Edit section: Roads"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Vehicle" title="Vehicle">vehicular</a> <a href="/wiki/Engineering" title="Engineering">engineering</a>, various <a href="/wiki/Landform" title="Landform">land</a>-based designs (<a href="/wiki/Automobiles" class="mw-redirect" title="Automobiles">automobiles</a>, <a href="/wiki/Sport_utility_vehicle" class="mw-redirect" title="Sport utility vehicle">sport utility vehicles</a>, <a href="/wiki/Truck" title="Truck">trucks</a>, <a href="/wiki/Train" title="Train">trains</a>, etc.) are rated for their ability to ascend <a href="/wiki/Terrain" title="Terrain">terrain</a>. Trains typically rate much lower than automobiles. The highest grade a vehicle can ascend while maintaining a particular speed is sometimes termed that vehicle's "gradeability" (or, less often, "grade ability"). The lateral slopes of a highway geometry are sometimes called <a href="/wiki/Cut_and_fill" title="Cut and fill">fills</a> or <a href="/wiki/Cut_(earthworks)" title="Cut (earthworks)">cuts</a> where these techniques have been used to create them. </p><p>In the United States, the maximum grade for federally funded highways is specified in a design table based on terrain and design speeds,<sup id="cite_ref-AASHTO_Geometry_7-0" class="reference"><a href="#cite_note-AASHTO_Geometry-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> with up to 6% generally allowed in mountainous areas and hilly urban areas with exceptions for up to 7% grades on mountainous roads with speed limits below 60 mph (95 km/h). </p><p>The steepest roads in the world according to the Guinness Book of World Records are <a href="/wiki/Baldwin_Street" title="Baldwin Street">Baldwin Street</a> in Dunedin, New Zealand, <a href="/wiki/Ffordd_Pen_Llech" title="Ffordd Pen Llech">Ffordd Pen Llech</a> in Harlech, Wales<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Canton_Avenue" title="Canton Avenue">Canton Avenue</a> in Pittsburgh, Pennsylvania.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The Guinness World Record once again lists <a href="/wiki/Baldwin_Street" title="Baldwin Street">Baldwin Street</a> as the steepest street in the world, with a 34.8% grade (1 in 2.87) after a successful appeal<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> against the ruling that handed the title, briefly, to <a href="/wiki/Ffordd_Pen_Llech" title="Ffordd Pen Llech">Ffordd Pen Llech</a>. </p><p>A number of streets elsewhere have steeper grades than those listed in the Guinness Book. Drawing on the U.S. National Elevation Dataset, <a href="/wiki/7x7_(magazine)" title="7x7 (magazine)">7x7 (magazine)</a> identified ten blocks of public streets in San Francisco open to vehicular traffic in the city with grades over 30 percent. The steepest at 41 percent is the block of Bradford Street above Tompkins Avenue in the <a href="/wiki/Bernal_Heights" class="mw-redirect" title="Bernal Heights">Bernal Heights</a> neighborhood.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/San_Francisco_Municipal_Railway" title="San Francisco Municipal Railway">San Francisco Municipal Railway</a> operates bus service among <a href="/wiki/List_of_hills_in_San_Francisco" title="List of hills in San Francisco">the city's hills</a>. The steepest grade for bus operations is 23.1% by the <i><a href="/wiki/List_of_San_Francisco_Municipal_Railway_lines" title="List of San Francisco Municipal Railway lines">67-Bernal Heights</a></i> on Alabama Street between Ripley and Esmeralda Streets.<sup id="cite_ref-Muni_12-0" class="reference"><a href="#cite_note-Muni-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>Likewise, the Pittsburgh Department of Engineering and Construction recorded a grade of 37% (20°) for Canton Avenue.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> The street has formed part of a bicycle race since 1983.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <ul class="gallery mw-gallery-packed"> <li class="gallerybox" style="width: 142px"> <div class="thumb" style="width: 140px;"><span typeof="mw:File"><a href="/wiki/File:Nederlands_verkeersbord_J6.svg" class="mw-file-description" title="10% slope warning sign, Netherlands"><img alt="10% slope warning sign, Netherlands" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Nederlands_verkeersbord_J6.svg/210px-Nederlands_verkeersbord_J6.svg.png" decoding="async" width="140" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Nederlands_verkeersbord_J6.svg/315px-Nederlands_verkeersbord_J6.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/13/Nederlands_verkeersbord_J6.svg/420px-Nederlands_verkeersbord_J6.svg.png 2x" data-file-width="350" data-file-height="300" /></a></span></div> <div class="gallerytext">10% slope warning sign, <a href="/wiki/Netherlands" title="Netherlands">Netherlands</a></div> </li> <li class="gallerybox" style="width: 138.66666666667px"> <div class="thumb" style="width: 136.66666666667px;"><span typeof="mw:File"><a href="/wiki/File:Finland_road_sign_115.svg" class="mw-file-description" title="7% descent warning sign, Finland"><img alt="7% descent warning sign, Finland" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Finland_road_sign_115.svg/205px-Finland_road_sign_115.svg.png" decoding="async" width="137" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Finland_road_sign_115.svg/308px-Finland_road_sign_115.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Finland_road_sign_115.svg/410px-Finland_road_sign_115.svg.png 2x" data-file-width="3014" data-file-height="2645" /></a></span></div> <div class="gallerytext">7% descent warning sign, <a href="/wiki/Finland" title="Finland">Finland</a></div> </li> <li class="gallerybox" style="width: 162px"> <div class="thumb" style="width: 160px;"><span typeof="mw:File"><a href="/wiki/File:Devil%27s_Staircase_Wales.jpg" class="mw-file-description" title="25% ascent warning sign, Wales"><img alt="25% ascent warning sign, Wales" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Devil%27s_Staircase_Wales.jpg/240px-Devil%27s_Staircase_Wales.jpg" decoding="async" width="160" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Devil%27s_Staircase_Wales.jpg/360px-Devil%27s_Staircase_Wales.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Devil%27s_Staircase_Wales.jpg/480px-Devil%27s_Staircase_Wales.jpg 2x" data-file-width="2272" data-file-height="1704" /></a></span></div> <div class="gallerytext">25% ascent warning sign, <a href="/wiki/Wales" title="Wales">Wales</a></div> </li> <li class="gallerybox" style="width: 92px"> <div class="thumb" style="width: 90px;"><span typeof="mw:File"><a href="/wiki/File:Pente30%25.jpg" class="mw-file-description" title="30% descent warning sign, over 1500 m. La Route des Crêtes, Cassis, France"><img alt="30% descent warning sign, over 1500 m. La Route des Crêtes, Cassis, France" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Pente30%25.jpg/135px-Pente30%25.jpg" decoding="async" width="90" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Pente30%25.jpg/202px-Pente30%25.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Pente30%25.jpg/270px-Pente30%25.jpg 2x" data-file-width="375" data-file-height="500" /></a></span></div> <div class="gallerytext">30% descent warning sign, over 1500 m. La Route des Crêtes, <a href="/wiki/Cassis" title="Cassis">Cassis</a>, <a href="/wiki/France" title="France">France</a></div> </li> <li class="gallerybox" style="width: 202px"> <div class="thumb" style="width: 200px;"><span typeof="mw:File"><a href="/wiki/File:Seattle_AM_General_trolleybus_climbing_James_St_near_5th_Ave_in_1983.jpg" class="mw-file-description" title="A trolleybus climbing an 18% grade in Seattle"><img alt="A trolleybus climbing an 18% grade in Seattle" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Seattle_AM_General_trolleybus_climbing_James_St_near_5th_Ave_in_1983.jpg/300px-Seattle_AM_General_trolleybus_climbing_James_St_near_5th_Ave_in_1983.jpg" decoding="async" width="200" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Seattle_AM_General_trolleybus_climbing_James_St_near_5th_Ave_in_1983.jpg/449px-Seattle_AM_General_trolleybus_climbing_James_St_near_5th_Ave_in_1983.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/75/Seattle_AM_General_trolleybus_climbing_James_St_near_5th_Ave_in_1983.jpg/599px-Seattle_AM_General_trolleybus_climbing_James_St_near_5th_Ave_in_1983.jpg 2x" data-file-width="2304" data-file-height="1386" /></a></span></div> <div class="gallerytext">A <a href="/wiki/Trolleybus" title="Trolleybus">trolleybus</a> climbing an 18% grade in <a href="/wiki/Seattle" title="Seattle">Seattle</a></div> </li> <li class="gallerybox" style="width: 144.66666666667px"> <div class="thumb" style="width: 142.66666666667px;"><span typeof="mw:File"><a href="/wiki/File:B10_Enzweihinger_Steige_20060528.JPG" class="mw-file-description" title="Ascent of German Bundesstraße 10"><img alt="Ascent of German Bundesstraße 10" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/B10_Enzweihinger_Steige_20060528.JPG/214px-B10_Enzweihinger_Steige_20060528.JPG" decoding="async" width="143" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/B10_Enzweihinger_Steige_20060528.JPG/321px-B10_Enzweihinger_Steige_20060528.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c7/B10_Enzweihinger_Steige_20060528.JPG/428px-B10_Enzweihinger_Steige_20060528.JPG 2x" data-file-width="5616" data-file-height="4729" /></a></span></div> <div class="gallerytext">Ascent of German <a href="/wiki/Bundesstra%C3%9Fe_10" title="Bundesstraße 10">Bundesstraße 10</a></div> </li> <li class="gallerybox" style="width: 162px"> <div class="thumb" style="width: 160px;"><span typeof="mw:File"><a href="/wiki/File:DunedinBaldwinStreet_Parked_Car.jpg" class="mw-file-description" title="A car parked on Baldwin Street, Dunedin, New Zealand"><img alt="A car parked on Baldwin Street, Dunedin, New Zealand" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/DunedinBaldwinStreet_Parked_Car.jpg/240px-DunedinBaldwinStreet_Parked_Car.jpg" decoding="async" width="160" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/DunedinBaldwinStreet_Parked_Car.jpg/360px-DunedinBaldwinStreet_Parked_Car.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b8/DunedinBaldwinStreet_Parked_Car.jpg/480px-DunedinBaldwinStreet_Parked_Car.jpg 2x" data-file-width="1024" data-file-height="768" /></a></span></div> <div class="gallerytext">A car parked on <a href="/wiki/Baldwin_Street" title="Baldwin Street">Baldwin Street</a>, <a href="/wiki/Dunedin,_New_Zealand" class="mw-redirect" title="Dunedin, New Zealand">Dunedin, New Zealand</a></div> </li> <li class="gallerybox" style="width: 162px"> <div class="thumb" style="width: 160px;"><span typeof="mw:File"><a href="/wiki/File:CantonAve_Top.jpg" class="mw-file-description" title="Looking down Canton Avenue, Pittsburgh, Pennsylvania"><img alt="Looking down Canton Avenue, Pittsburgh, Pennsylvania" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/CantonAve_Top.jpg/240px-CantonAve_Top.jpg" decoding="async" width="160" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/CantonAve_Top.jpg/360px-CantonAve_Top.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/34/CantonAve_Top.jpg/480px-CantonAve_Top.jpg 2x" data-file-width="1024" data-file-height="768" /></a></span></div> <div class="gallerytext">Looking down <a href="/wiki/Canton_Avenue" title="Canton Avenue">Canton Avenue</a>, <a href="/wiki/Pittsburgh,_Pennsylvania" class="mw-redirect" title="Pittsburgh, Pennsylvania">Pittsburgh, Pennsylvania</a></div> </li> </ul> <div class="mw-heading mw-heading2"><h2 id="Environmental_design">Environmental design</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grade_(slope)&action=edit&section=7" title="Edit section: Environmental design"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Grade, pitch, and slope are important components in <a href="/wiki/Landscape_design" title="Landscape design">landscape design</a>, <a href="/wiki/Garden_design" title="Garden design">garden design</a>, <a href="/wiki/Landscape_architecture" title="Landscape architecture">landscape architecture</a>, and <a href="/wiki/Architecture" title="Architecture">architecture</a>; for engineering and aesthetic design factors. Drainage, slope stability, circulation of people and vehicles, complying with building codes, and design integration are all aspects of slope considerations in <a href="/wiki/Environmental_design" title="Environmental design">environmental design</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Railways">Railways</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grade_(slope)&action=edit&section=8" title="Edit section: Railways"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="anchor" id="Railroad_gradient"></span> </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Track_Grade_indicator_150-88.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Track_Grade_indicator_150-88.JPG/220px-Track_Grade_indicator_150-88.JPG" decoding="async" width="220" height="157" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Track_Grade_indicator_150-88.JPG/330px-Track_Grade_indicator_150-88.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Track_Grade_indicator_150-88.JPG/440px-Track_Grade_indicator_150-88.JPG 2x" data-file-width="1280" data-file-height="916" /></a><figcaption>Grade indicator near <a href="/wiki/Bellville,_Western_Cape" class="mw-redirect" title="Bellville, Western Cape">Bellville, Western Cape</a>, <a href="/wiki/South_Africa" title="South Africa">South Africa</a>, showing 1:150 and 1:88 grades.</figcaption></figure> <p><a href="/wiki/Ruling_gradient" title="Ruling gradient">Ruling gradients</a> limit the load that a <a href="/wiki/Locomotive" title="Locomotive">locomotive</a> can haul, including the weight of the locomotive itself. Gradients are expressed as a ratio of vertical rise to horizontal distance; for example, a 1% gradient (1 in 100) means the track rises 1 vertical unit for every 100 horizontal units. On such a gradient, a locomotive can pull half (or less) of the load that it can pull on level track. (A heavily loaded train rolling at 20 km/h on heavy rail may require ten times the pull on a 1% upgrade that it does on the level at that speed.) </p><p>Early railways in the United Kingdom were laid out with very gentle gradients, such as 0.07575% (1 in 1320) and 0.1515% (1 in 660) on the <a href="/wiki/Great_Western_main_line" class="mw-redirect" title="Great Western main line">Great Western main line</a>, nicknamed <a href="/wiki/Isambard_Kingdom_Brunel" title="Isambard Kingdom Brunel">Brunel's</a> Billiard Table, because the early locomotives (and their brakes) were feeble. Steep gradients were concentrated in short sections of lines where it was convenient to employ <a href="/wiki/Bank_engine" title="Bank engine">assistant engines</a> or <a href="/wiki/Cable_railway" title="Cable railway">cable haulage</a>, such as the 1.2 kilometres (0.75 miles) section from <a href="/wiki/Euston_railway_station" title="Euston railway station">Euston</a> to <a href="/wiki/Camden_Town" title="Camden Town">Camden Town</a>. </p><p>Extremely steep gradients require the use of cables (such as the Scenic Railway at <a href="/wiki/Katoomba_Scenic_World" class="mw-redirect" title="Katoomba Scenic World">Katoomba Scenic World</a>, Australia, with a maximum grade of 122% (52°), claimed to be the world's steepest passenger-carrying <a href="/wiki/Funicular" title="Funicular">funicular</a><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup>) or some kind of <a href="/wiki/Rack_railway" title="Rack railway">rack railway</a> (such as the <a href="/wiki/Pilatus_railway" class="mw-redirect" title="Pilatus railway">Pilatus railway</a> in Switzerland, with a maximum grade of 48% (26°), claimed to be the world's steepest rack railway<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup>) to help the train ascend or descend. </p><p>Gradients can be expressed as an angle, as feet per mile, feet per chain, 1 in <span class="texhtml mvar" style="font-style:italic;">n</span>, <span class="texhtml mvar" style="font-style:italic;">x</span>% or <span class="texhtml mvar" style="font-style:italic;">y</span> per mille. Since designers like round figures, the method of expression can affect the gradients selected.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (May 2021)">citation needed</span></a></i>]</sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Sklon%C3%ADk-kles%C3%A1n%C3%AD.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Sklon%C3%ADk-kles%C3%A1n%C3%AD.jpg/220px-Sklon%C3%ADk-kles%C3%A1n%C3%AD.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Sklon%C3%ADk-kles%C3%A1n%C3%AD.jpg/330px-Sklon%C3%ADk-kles%C3%A1n%C3%AD.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Sklon%C3%ADk-kles%C3%A1n%C3%AD.jpg/440px-Sklon%C3%ADk-kles%C3%A1n%C3%AD.jpg 2x" data-file-width="2048" data-file-height="1536" /></a><figcaption>A 1371-metre long stretch of railroad with a 20<a href="/wiki/Permille" class="mw-redirect" title="Permille">‰</a> (2%) slope, <a href="/wiki/Czech_Republic" title="Czech Republic">Czech Republic</a></figcaption></figure> <p>The steepest <a href="/wiki/Rail_adhesion" class="mw-redirect" title="Rail adhesion">railway lines</a> that do not use a rack system include: </p> <ul><li>13.5% (1 in 7.40) – <a href="/wiki/Trams_in_Lisbon" title="Trams in Lisbon">Lisbon tram</a>, Portugal</li> <li>11.6% (1 in 8.62) – <a href="/wiki/P%C3%B6stlingbergbahn" title="Pöstlingbergbahn">Pöstlingbergbahn</a>, <a href="/wiki/Linz" title="Linz">Linz</a>, Austria<sup id="cite_ref-2009brochure_17-0" class="reference"><a href="#cite_note-2009brochure-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup></li> <li>11.0% (1 in 9.09) – <a href="/wiki/Cass_Scenic_Railway" class="mw-redirect" title="Cass Scenic Railway">Cass Scenic Railway</a>, US (former logging line)</li> <li>9.0% (1 in 11.11) – <a href="/wiki/Ligne_de_Saint_Gervais_%E2%80%93_Vallorcine" class="mw-redirect" title="Ligne de Saint Gervais – Vallorcine">Ligne de Saint Gervais – Vallorcine</a>, France</li> <li>9.0% (1 in 11.11) – <a href="/wiki/Muni_Metro" title="Muni Metro">Muni Metro</a> <a href="/wiki/J_Church" title="J Church">J Church</a>, <a href="/wiki/San_Francisco" title="San Francisco">San Francisco</a>, US<sup id="cite_ref-Muni_12-1" class="reference"><a href="#cite_note-Muni-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup></li> <li>8.8% (1 in 11.4) – <a href="/wiki/Trams_in_Ia%C8%99i" class="mw-redirect" title="Trams in Iași">Iași tram</a>, Romania<sup id="cite_ref-Iasi_18-0" class="reference"><a href="#cite_note-Iasi-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup></li> <li>8.65% (1 in 11.95) – <a href="/wiki/Portland_Streetcar" title="Portland Streetcar">Portland Streetcar</a>, <a href="/wiki/Oregon" title="Oregon">Oregon</a>, US<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup></li> <li>8.33%(1 in 12) – <a href="/wiki/Nilgiri_Mountain_Railway" title="Nilgiri Mountain Railway">Nilgiri Mountain Railway</a> Tamil Nadu, India</li> <li>8.0% (1 in 12.5) – Just outside the Tombstone Jct. Station in the <a href="/wiki/Tombstone_Junction" title="Tombstone Junction">Tombstone Junction</a> Theme Park, <a href="/wiki/Kentucky" title="Kentucky">Kentucky</a>, US. The railroad line there had a <a href="/wiki/Ruling_grade" class="mw-redirect" title="Ruling grade">ruling grade</a> of 6% (1 in 16.7).</li> <li>7.85% (1 in 12.7) – Near <a href="/wiki/Washington_Street_station_(MBTA)" title="Washington Street station (MBTA)">Washington Street station</a> on the <a href="/wiki/Green_Line_B_branch" title="Green Line B branch">Green Line B branch</a>, <a href="/wiki/Boston" title="Boston">Boston</a>, <a href="/wiki/Massachusetts" title="Massachusetts">Massachusetts</a>, US.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup></li> <li>7.1% (1 in 14.08) – <a href="/wiki/Schneeberg_Railway_(cog_railway)#Austrian_rack_railways_no_longer_operating" title="Schneeberg Railway (cog railway)">Erzberg Railway</a>, Austria</li> <li>7.0% (1 in 14.28) – <a href="/wiki/Bernina_Railway" class="mw-redirect" title="Bernina Railway">Bernina Railway</a>, Switzerland</li> <li>6.0% (1 in 16.7) – <a href="/wiki/Arica" title="Arica">Arica</a>, Chile to <a href="/wiki/La_Paz" title="La Paz">La Paz</a>, Bolivia</li> <li>6.0% (1 in 16.6) – <a href="/wiki/Docklands_Light_Railway" title="Docklands Light Railway">Docklands Light Railway</a>, London, UK</li> <li>6.0% (1 in 16.6) – <a href="/wiki/Ferrovia_Centrale_Umbra" class="mw-redirect" title="Ferrovia Centrale Umbra">Ferrovia Centrale Umbra</a>, Italy<sup id="cite_ref-cifi18_21-0" class="reference"><a href="#cite_note-cifi18-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup></li> <li>6.0% (1 in 16.6) - <a href="/wiki/Link_light_rail" title="Link light rail">Link light rail</a>, <a href="/wiki/Seattle" title="Seattle">Seattle</a>, U.S.</li> <li>5.89% (1 in 16.97) – <a href="/wiki/Madison,_Indiana" title="Madison, Indiana">Madison, Indiana</a>, US<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup></li> <li>5.6% (1 in 18) – <a href="/wiki/Fl%C3%A5m_Line" title="Flåm Line">Flåm Line</a>, Norway</li> <li>5.3% (1 in 19) – <a href="/wiki/Foxfield_Railway" title="Foxfield Railway">Foxfield Railway</a>, <a href="/wiki/Staffordshire" title="Staffordshire">Staffordshire</a>, UK</li> <li>5.1% (1 in 19.6) – <a href="/wiki/Saluda_Grade" title="Saluda Grade">Saluda Grade</a>, <a href="/wiki/North_Carolina" title="North Carolina">North Carolina</a>, US</li> <li>5.0% (1 in 20) – <a href="/wiki/Khyber_Pass_Railway" title="Khyber Pass Railway">Khyber Pass Railway</a>, Pakistan</li> <li>4.5% (1 in 22.2) – The <a href="/wiki/Canadian_Pacific_Railway" title="Canadian Pacific Railway">Canadian Pacific Railway</a>'s <a href="/wiki/Big_Hill" title="Big Hill">Big Hill</a>, <a href="/wiki/British_Columbia" title="British Columbia">British Columbia</a>, Canada (prior to the construction of the <a href="/wiki/Big_Hill#The_Spiral_Tunnels" title="Big Hill">Spiral Tunnels</a>)</li> <li>4.3% (1 in 23) – <a href="/wiki/Colwall_Tunnels" title="Colwall Tunnels">Colwall Old Tunnel</a>, <a href="/wiki/Malvern_Hills" title="Malvern Hills">Malvern Hills</a>, UK<sup id="cite_ref-Colwall_23-0" class="reference"><a href="#cite_note-Colwall-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup></li> <li>4.0% (1 in 25) – <a href="/wiki/Cologne-Frankfurt_high-speed_rail_line" class="mw-redirect" title="Cologne-Frankfurt high-speed rail line">Cologne-Frankfurt high-speed rail line</a>, Germany</li> <li>4.0% (1 in 25) – <a href="/wiki/Bolan_Pass" title="Bolan Pass">Bolan Pass Railway</a>, Pakistan</li> <li>4.0% (1 in 25) – (211.2 feet (64 m) per 1 mile (1,600 m) ) – <a href="/wiki/Oberon_railway_line" title="Oberon railway line">Tarana – Oberon branch</a>, <a href="/wiki/New_South_Wales" title="New South Wales">New South Wales</a>, Australia.</li> <li>4.0% (1 in 25) – <a href="/wiki/Matheran_Light_Railway" class="mw-redirect" title="Matheran Light Railway">Matheran Light Railway</a>, India<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup></li> <li>4.0% (1 in 26) – <a href="/wiki/Rewanui_Branch" title="Rewanui Branch">Rewanui Incline</a>, New Zealand. Fitted with Fell center rail but was not used for motive power, but only braking</li> <li>3.6% (1 in 27) – <a href="/wiki/Ecclesbourne_Valley_Railway" title="Ecclesbourne Valley Railway">Ecclesbourne Valley Railway</a>, Heritage Line, <a href="/wiki/Wirksworth" title="Wirksworth">Wirksworth</a>, <a href="/wiki/Derbyshire" title="Derbyshire">Derbyshire</a>, UK</li> <li>3.6% (1 in 28) – The Westmere Bank, New Zealand has a ruling gradient of 1 in 35, however peaks at 1 in 28</li> <li>3.33% (1 in 30) – <a href="/wiki/Umgeni_Steam_Railway" title="Umgeni Steam Railway">Umgeni Steam Railway</a>, <a href="/wiki/South_Africa" title="South Africa">South Africa</a><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup></li> <li>3.0% (1 in 33) – several sections of the <a href="/wiki/Main_Western_railway_line" class="mw-redirect" title="Main Western railway line">Main Western line</a> between <a href="/wiki/Valley_Heights_railway_station" title="Valley Heights railway station">Valley Heights</a> and <a href="/wiki/Katoomba_railway_station" title="Katoomba railway station">Katoomba</a> in the <a href="/wiki/Blue_Mountains,_New_South_Wales" class="mw-redirect" title="Blue Mountains, New South Wales">Blue Mountains</a> Australia.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup></li> <li>3.0% (1 in 33) – The entire <a href="/wiki/Newmarket_Line" title="Newmarket Line">Newmarket Line</a> in central Auckland, New Zealand</li> <li>3.0% (1 in 33) – <a href="/wiki/Otira_Tunnel" title="Otira Tunnel">Otira Tunnel</a>, New Zealand, which is equipped with extraction fans to reduce chance of overheating and low visibility</li> <li>3.0% (1 in 33) – The approaches to the George L. Anderson Memorial Bridge crossing the <a href="/wiki/Neponset_River" title="Neponset River">Neponset River</a>, Boston, Massachusetts, US. The <a href="/wiki/Ruling_grade" class="mw-redirect" title="Ruling grade">Ruling Gradient</a> of the Braintree Branch of the <a href="/wiki/Red_Line_(MBTA)" title="Red Line (MBTA)">Red Line (MBTA)</a>.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup></li> <li>2.7% (1 in 37) – <a href="/wiki/Braganza_Ghats" title="Braganza Ghats">Braganza Ghats</a>, <a href="/wiki/Bhor_Ghat" title="Bhor Ghat">Bhor Ghat</a> and <a href="/wiki/Kasara_ghat" class="mw-redirect" title="Kasara ghat">Thull ghat</a> sections in Indian Railways, India</li> <li>2.7% (1 in 37) – <a href="/wiki/Exeter_Central_railway_station" title="Exeter Central railway station">Exeter Central</a> to <a href="/wiki/Exeter_St_Davids_railway_station" title="Exeter St Davids railway station">Exeter St Davids</a>, UK (see <a href="/wiki/Exeter_Central_railway_station#Description" title="Exeter Central railway station">Exeter Central railway station#Description</a>)</li> <li>2.7% (1 in 37) – Picton- Elevation, New Zealand</li> <li>2.65% (1 in 37.7) – <a href="/wiki/Lickey_Incline" title="Lickey Incline">Lickey Incline</a>, UK</li> <li>2.6% (1 in 38) – A slope near <a href="/wiki/Halden" title="Halden">Halden</a> on <a href="/wiki/%C3%98stfold_Line" title="Østfold Line">Østfold Line</a>, Norway – Ok for passenger <a href="/wiki/Multiple_units" class="mw-redirect" title="Multiple units">multiple units</a>, but an obstacle for freight trains which must keep their weight down on this international mainline because of the slope. Freight traffic has mainly shifted to road.</li> <li>2.3% (1 in 43.5) – <a href="/wiki/Schiefe_Ebene" title="Schiefe Ebene">Schiefe Ebene</a>, Germany</li> <li>2.2% (1 in 45.5) – The <a href="/wiki/Canadian_Pacific_Railway" title="Canadian Pacific Railway">Canadian Pacific Railway</a>'s <a href="/wiki/Big_Hill" title="Big Hill">Big Hill</a>, <a href="/wiki/British_Columbia" title="British Columbia">British Columbia</a>, Canada (after the construction of the <a href="/wiki/Big_Hill#The_Spiral_Tunnels" title="Big Hill">Spiral Tunnels</a>)</li> <li>2.0% (1 in 48) – Beasdale Bank (West Coast Scotland mainline), UK</li> <li>2.0% (1 in 50) – Numerous locations on New Zealand's railway network, New Zealand</li> <li>1.66% (1 in 60) – Section of track on the <a href="/wiki/Dublin%E2%80%93Cork_railway_line" title="Dublin–Cork railway line">Dublin-Cork line</a> between mile posts 164 and 162.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup></li> <li>1.51% (1 in 66) – (1 foot (0.3 m) per 1 chain (20 m)) <a href="/wiki/New_South_Wales_Government_Railways" title="New South Wales Government Railways">New South Wales Government Railways</a>, Australia, part of <a href="/wiki/Main_Southern_railway_line" class="mw-redirect" title="Main Southern railway line">Main South line</a>.</li> <li>1.25% (1 in 80) – <a href="/wiki/Wellington_Bank,_Somerset" title="Wellington Bank, Somerset">Wellington Bank, Somerset</a>, UK</li> <li>1.25% (1 in 80) – <a href="/wiki/Rudgwick_railway_station" title="Rudgwick railway station">Rudgwick</a>, UK (<a href="/wiki/West_Sussex" title="West Sussex">West Sussex</a>) platform before regrading – too steep if a train is not provided with <a href="/wiki/Railway_brake#Continuous_brakes" title="Railway brake">continuous brakes</a>.</li> <li>0.77% (1 in 130) – <a href="/wiki/Rudgwick_railway_station" title="Rudgwick railway station">Rudgwick</a>, UK platform after regrading – not too steep if a train is not provided with continuous brakes.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Compensation_for_curvature">Compensation for curvature</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grade_(slope)&action=edit&section=9" title="Edit section: Compensation for curvature"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Gradients on sharp curves are effectively a bit steeper than the same gradient on straight track, so to compensate for this and make the <a href="/wiki/Ruling_grade" class="mw-redirect" title="Ruling grade">ruling grade</a> uniform throughout, the gradient on those sharp curves should be reduced slightly. </p> <div class="mw-heading mw-heading3"><h3 id="Continuous_brakes">Continuous brakes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grade_(slope)&action=edit&section=10" title="Edit section: Continuous brakes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the era before they were provided with <a href="/wiki/Railway_brake#Continuous_brakes" title="Railway brake">continuous brakes</a>, whether <a href="/wiki/Railway_air_brake" title="Railway air brake">air brakes</a> or <a href="/wiki/Vacuum_brake" title="Vacuum brake">vacuum brakes</a>, steep gradients made it extremely difficult for trains to stop safely. In those days, for example, an <a href="/wiki/Her_Majesty%27s_Railway_Inspectorate" class="mw-redirect" title="Her Majesty's Railway Inspectorate">inspector</a> insisted that <a href="/wiki/Rudgwick_railway_station" title="Rudgwick railway station">Rudgwick railway station</a> in <a href="/wiki/West_Sussex" title="West Sussex">West Sussex</a> be regraded. He would not allow it to open until the gradient through the platform was eased from 1 in 80 to 1 in 130. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grade_(slope)&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 25em;"> <ul><li><a href="/wiki/Angle_of_repose" title="Angle of repose">Angle of repose</a></li> <li><a href="/wiki/Aspect_(geography)" title="Aspect (geography)">Aspect (geography)</a></li> <li><a href="/wiki/Civil_engineering" title="Civil engineering">Civil engineering</a></li> <li><a href="/wiki/Construction_surveying" title="Construction surveying">Construction surveying</a></li> <li><a href="/wiki/Grading_(engineering)" class="mw-redirect" title="Grading (engineering)">Grading (engineering)</a></li> <li><a href="/wiki/Tunnel#Cut-and-cover" title="Tunnel">Cut-and-cover</a></li> <li><a href="/wiki/Cut_(earthmoving)" class="mw-redirect" title="Cut (earthmoving)">Cut (earthmoving)</a></li> <li><a href="/wiki/Cut_and_fill" title="Cut and fill">Cut and fill</a></li> <li><a href="/wiki/Embankment_(transportation)" class="mw-redirect" title="Embankment (transportation)">Embankment (transportation)</a></li> <li><a href="/wiki/Grade_separation" title="Grade separation">Grade separation</a></li> <li><a href="/wiki/Inclined_plane" title="Inclined plane">Inclined plane</a></li> <li><a href="/wiki/List_of_steepest_gradients_on_adhesion_railways" title="List of steepest gradients on adhesion railways">List of steepest gradients on adhesion railways</a></li> <li><a href="/wiki/Per_mille" title="Per mille">Per mille</a></li> <li><a href="/wiki/Percentage#Other_uses" title="Percentage">Percentage</a></li> <li><a href="/wiki/Rake_(theatre)" title="Rake (theatre)">Rake</a></li> <li><a href="/wiki/Roof_pitch" title="Roof pitch">Roof pitch</a></li> <li><a href="/wiki/Ruling_gradient" title="Ruling gradient">Ruling gradient</a></li> <li><a href="/wiki/Slope_stability" title="Slope stability">Slope stability</a></li> <li><a href="/wiki/Slope" title="Slope">Slope</a> <ul><li><a href="/wiki/Slope_stability_analysis" title="Slope stability analysis">Slope stability analysis</a></li></ul></li> <li><a href="/wiki/Stream_slope" class="mw-redirect" title="Stream slope">Stream slope</a></li> <li><a href="/wiki/Strike_and_dip" title="Strike and dip">Strike and dip</a></li> <li><a href="/wiki/Surface_gradient" title="Surface gradient">Surface gradient</a></li> <li><a href="/wiki/Surveying" title="Surveying">Surveying</a></li> <li><a href="/wiki/Trench" title="Trench">Trench</a></li> <li><a href="/wiki/Tunnel" title="Tunnel">Tunnel</a></li> <li><a href="/wiki/Wheelchair_ramp" title="Wheelchair ramp">Wheelchair ramp</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grade_(slope)&action=edit&section=12" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 25em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFMichael_W._SayersThomas_D._GillespieWilliam_D.O._Paterson1986" class="citation web cs1 cs1-prop-foreign-lang-source">Michael W. Sayers; Thomas D. Gillespie; William D.O. Paterson (January 1986). <a rel="nofollow" class="external text" href="https://documents1.worldbank.org/curated/en/864001468334203776/pdf/WTP460PUB0FREN0s0Appareils000French.pdf">"Directives pour la mesure de l'uni des routes et l'étalonnage des appareils"</a> <span class="cs1-format">(PDF)</span> (in French). <a href="/wiki/World_Bank" title="World Bank">World Bank</a>. p. 2. <q>L'indice IRI est une mesure de l'uni des routes standardisée, apparentée aux mesures obtenues à l'aide des appareils de type-réponse. 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Collegio Ingegneri Ferroviari Italiani. 15 October 2018<span class="reference-accessdate">. Retrieved <span class="nowrap">23 May</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Il+Piano+Tecnologico+di+RFI&rft.pub=Collegio+Ingegneri+Ferroviari+Italiani&rft.date=2018-10-15&rft_id=http%3A%2F%2Fwww.cifi.it%2FUplDocumenti%2FFirenze15102018%2F7%2520Ing.Rosini.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGrade+%28slope%29" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.oldmadison.com/madview2.html">"Madisonview"</a>. <i>oldmadison.com</i><span class="reference-accessdate">. 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UNESCO World Heritage Convention.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Matheran+Light+Railway+%28extension+to+the+Mountain+Railways+of+India%29&rft.pub=UNESCO+World+Heritage+Convention&rft_id=https%3A%2F%2Fwhc.unesco.org%2Fen%2Ftentativelists%2F2082%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGrade+%28slope%29" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMartin2005" class="citation web cs1">Martin, Bruno (September 2005). <a rel="nofollow" class="external text" href="http://www.sa-transport.co.za/trains/maps/NML-Dbn-Pmb.pdf">"Durban-Pietermaritzburg main line map and profile"</a> <span class="cs1-format">(PDF)</span>. <i>Transport in South and Southern Africa</i><span class="reference-accessdate">. Retrieved <span class="nowrap">7 April</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Transport+in+South+and+Southern+Africa&rft.atitle=Durban-Pietermaritzburg+main+line+map+and+profile&rft.date=2005-09&rft.aulast=Martin&rft.aufirst=Bruno&rft_id=http%3A%2F%2Fwww.sa-transport.co.za%2Ftrains%2Fmaps%2FNML-Dbn-Pmb.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGrade+%28slope%29" class="Z3988"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><a href="/wiki/Valley_Heights_railway_station" title="Valley Heights railway station">Valley Heights railway station</a></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.ctps.org/data/pdf/studies/highway/se_expr_concept.pdf">"Improving the Southeast Expressway: A Conceptual Plan"</a> <span class="cs1-format">(PDF)</span>. <i>bostonmpo.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">24 February</span> 2021</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=bostonmpo.org&rft.atitle=Improving+the+Southeast+Expressway%3A+A+Conceptual+Plan&rft_id=https%3A%2F%2Fwww.ctps.org%2Fdata%2Fpdf%2Fstudies%2Fhighway%2Fse_expr_concept.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGrade+%28slope%29" class="Z3988"></span></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.irishrail.ie/getmedia/3ec0febe-9be1-43a7-92ae-04e965146165/IE-2023-Network-Statement-updated-29-04-2024.pdf">"Network Statement 2023"</a> <span class="cs1-format">(PDF)</span>. Irish Rail.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Network+Statement+2023&rft.pub=Irish+Rail&rft_id=https%3A%2F%2Fwww.irishrail.ie%2Fgetmedia%2F3ec0febe-9be1-43a7-92ae-04e965146165%2FIE-2023-Network-Statement-updated-29-04-2024.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGrade+%28slope%29" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grade_(slope)&action=edit&section=13" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1 cs1-prop-unfit"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20171016174532/http://www.railsigns.uk/sect24page1/sect24page1.html">"British railway gradients and their signs"</a>. <i>Railsigns</i>. 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template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Geotechnical_engineering" title="Special:EditPage/Template:Geotechnical engineering"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Geotechnical_engineering316" style="font-size:114%;margin:0 4em"><a href="/wiki/Geotechnical_engineering" title="Geotechnical engineering">Geotechnical engineering</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div><a href="/wiki/Offshore_geotechnical_engineering" title="Offshore geotechnical engineering">Offshore geotechnical engineering</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Geotechnical_investigation" title="Geotechnical investigation">Investigation</a> <br />and<br />instrumentation</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:10.0em;font-weight: normal;">Field (<i>in situ</i>)</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"><div class="div-col" style="column-width: 33em;"> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Core_sample.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Core_sample.svg/20px-Core_sample.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Core_sample.svg/30px-Core_sample.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Core_sample.svg/40px-Core_sample.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/wiki/Core_drill" title="Core drill">Core drill</a></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Cone_penetration_test.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Cone_penetration_test.svg/20px-Cone_penetration_test.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Cone_penetration_test.svg/30px-Cone_penetration_test.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/28/Cone_penetration_test.svg/40px-Cone_penetration_test.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/wiki/Cone_penetration_test" title="Cone penetration test">Cone penetration test</a></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Geo-electrical_sounding.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Geo-electrical_sounding.svg/20px-Geo-electrical_sounding.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Geo-electrical_sounding.svg/30px-Geo-electrical_sounding.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/10/Geo-electrical_sounding.svg/40px-Geo-electrical_sounding.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/w/index.php?title=Geo-electrical_sounding&action=edit&redlink=1" class="new" title="Geo-electrical sounding (page does not exist)">Geo-electrical sounding</a></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:In_situ_permeameter_test.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/In_situ_permeameter_test.svg/20px-In_situ_permeameter_test.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/In_situ_permeameter_test.svg/30px-In_situ_permeameter_test.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7b/In_situ_permeameter_test.svg/40px-In_situ_permeameter_test.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/wiki/Permeability_(Earth_sciences)" class="mw-redirect" title="Permeability (Earth sciences)">Permeability test</a></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Load_test.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Load_test.svg/20px-Load_test.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Load_test.svg/30px-Load_test.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Load_test.svg/40px-Load_test.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/wiki/Load_test" class="mw-redirect" title="Load test">Load test</a> <ul><li><a href="/wiki/Static_load_testing" title="Static load testing">Static</a></li> <li><a href="/wiki/Dynamic_load_testing" title="Dynamic load testing">Dynamic</a></li> <li><a href="/wiki/Statnamic_load_test" title="Statnamic load test">Statnamic</a></li></ul></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Pore_pressure_measurement.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Pore_pressure_measurement.svg/20px-Pore_pressure_measurement.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Pore_pressure_measurement.svg/30px-Pore_pressure_measurement.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Pore_pressure_measurement.svg/40px-Pore_pressure_measurement.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> Pore pressure measurement <ul><li><a href="/wiki/Piezometer" class="mw-redirect" title="Piezometer">Piezometer</a></li> <li><a href="/wiki/Well#Classification" title="Well">Well</a></li></ul></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Ram_sounding.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Ram_sounding.svg/20px-Ram_sounding.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Ram_sounding.svg/30px-Ram_sounding.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Ram_sounding.svg/40px-Ram_sounding.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/w/index.php?title=Ram_sounding&action=edit&redlink=1" class="new" title="Ram sounding (page does not exist)">Ram sounding</a></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Rock_control_drilling.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Rock_control_drilling.svg/20px-Rock_control_drilling.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Rock_control_drilling.svg/30px-Rock_control_drilling.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Rock_control_drilling.svg/40px-Rock_control_drilling.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/w/index.php?title=Rock_control_drilling&action=edit&redlink=1" class="new" title="Rock control drilling (page does not exist)">Rock control drilling</a></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Rotary_pressure_sounding.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Rotary_pressure_sounding.svg/20px-Rotary_pressure_sounding.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Rotary_pressure_sounding.svg/30px-Rotary_pressure_sounding.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/81/Rotary_pressure_sounding.svg/40px-Rotary_pressure_sounding.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/wiki/Rotary-pressure_sounding" title="Rotary-pressure sounding">Rotary-pressure sounding</a></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Rotary_weight_sounding.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Rotary_weight_sounding.svg/20px-Rotary_weight_sounding.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Rotary_weight_sounding.svg/30px-Rotary_weight_sounding.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Rotary_weight_sounding.svg/40px-Rotary_weight_sounding.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/w/index.php?title=Rotary_weight_sounding&action=edit&redlink=1" class="new" title="Rotary weight sounding (page does not exist)">Rotary weight sounding</a></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Sample_series.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Sample_series.svg/20px-Sample_series.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Sample_series.svg/30px-Sample_series.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/05/Sample_series.svg/40px-Sample_series.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/wiki/Soil_test#Soil_testing" title="Soil test">Sample series</a></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Screw_plate_test.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Screw_plate_test.svg/20px-Screw_plate_test.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Screw_plate_test.svg/30px-Screw_plate_test.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/75/Screw_plate_test.svg/40px-Screw_plate_test.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/w/index.php?title=Screw_plate_test&action=edit&redlink=1" class="new" title="Screw plate test (page does not exist)">Screw plate test</a></li></ul> <ul><li><a href="/wiki/Deformation_monitoring" title="Deformation monitoring">Deformation monitoring</a> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Inclinometer.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Inclinometer.svg/20px-Inclinometer.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Inclinometer.svg/30px-Inclinometer.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Inclinometer.svg/40px-Inclinometer.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/wiki/Inclinometer" title="Inclinometer">Inclinometer</a></li> <li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Settlement_recordings.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Settlement_recordings.svg/20px-Settlement_recordings.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Settlement_recordings.svg/30px-Settlement_recordings.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/35/Settlement_recordings.svg/40px-Settlement_recordings.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/wiki/Soil_consolidation" title="Soil consolidation">Settlement recordings</a></li></ul></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Shear_vane_test.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Shear_vane_test.svg/20px-Shear_vane_test.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Shear_vane_test.svg/30px-Shear_vane_test.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Shear_vane_test.svg/40px-Shear_vane_test.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/wiki/Shear_vane_test" title="Shear vane test">Shear vane test</a></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Simple_sounding.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Simple_sounding.svg/20px-Simple_sounding.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Simple_sounding.svg/30px-Simple_sounding.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Simple_sounding.svg/40px-Simple_sounding.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/w/index.php?title=Simple_sounding&action=edit&redlink=1" class="new" title="Simple sounding (page does not exist)">Simple sounding</a></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Standard_penetration_test.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Standard_penetration_test.svg/20px-Standard_penetration_test.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Standard_penetration_test.svg/30px-Standard_penetration_test.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Standard_penetration_test.svg/40px-Standard_penetration_test.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/wiki/Standard_penetration_test" title="Standard penetration test">Standard penetration test</a></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Total_sounding.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Total_sounding.svg/20px-Total_sounding.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Total_sounding.svg/30px-Total_sounding.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/Total_sounding.svg/40px-Total_sounding.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/wiki/Total_sounding" title="Total sounding">Total sounding</a></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Trial_pit.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Trial_pit.svg/20px-Trial_pit.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Trial_pit.svg/30px-Trial_pit.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/71/Trial_pit.svg/40px-Trial_pit.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/wiki/Trial_pit" title="Trial pit">Trial pit</a></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Visible_rock.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Visible_rock.svg/20px-Visible_rock.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Visible_rock.svg/30px-Visible_rock.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/14/Visible_rock.svg/40px-Visible_rock.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/wiki/Bedrock" title="Bedrock">Visible bedrock</a></li></ul> <ul><li><a href="/wiki/Nuclear_densometer" class="mw-redirect" title="Nuclear densometer">Nuclear densometer test</a></li></ul> <ul><li><a href="/wiki/Exploration_geophysics" title="Exploration geophysics">Exploration geophysics</a></li></ul> <ul><li><a href="/wiki/Crosshole_sonic_logging" title="Crosshole sonic logging">Crosshole sonic logging</a></li></ul> <ul><li><a href="/wiki/Pile_integrity_test" title="Pile integrity test">Pile integrity test</a></li></ul> <ul><li><a href="/wiki/Wave_equation_analysis" title="Wave equation analysis">Wave equation analysis</a></li></ul></div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:10.0em;font-weight: normal;"><a href="/wiki/Soil_test" title="Soil test">Laboratory <br />testing</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <li><a href="/wiki/Soil_classification" title="Soil classification">Soil classification</a></li> <li><a href="/wiki/Atterberg_limits" title="Atterberg limits">Atterberg limits</a></li> <li><a href="/wiki/California_bearing_ratio" title="California bearing ratio">California bearing ratio</a></li> <li><a href="/wiki/Direct_shear_test" title="Direct shear test">Direct shear test</a></li> <li><a href="/wiki/Hydrometer" title="Hydrometer">Hydrometer</a></li> <li><a href="/wiki/Proctor_compaction_test" title="Proctor compaction test">Proctor compaction test</a></li> <li><a href="/wiki/R-value_(soils)" title="R-value (soils)">R-value</a></li> <li><a href="/wiki/Sieve_analysis" title="Sieve analysis">Sieve analysis</a></li> <li><a href="/wiki/Triaxial_shear_test" title="Triaxial shear test">Triaxial shear test</a></li> <li><a href="/wiki/Oedometer_test" title="Oedometer test">Oedometer test</a></li> <li><a href="/wiki/Hydraulic_conductivity#Experimental_approach" title="Hydraulic conductivity">Hydraulic conductivity tests</a></li> <li><a href="/wiki/Water_content#Measurement" title="Water content">Water content tests</a></li> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Soil" title="Soil">Soil</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:10.0em;font-weight: normal;">Types</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Clay" title="Clay">Clay</a></li> <li><a href="/wiki/Silt" title="Silt">Silt</a></li> <li><a href="/wiki/Sand" title="Sand">Sand</a></li> <li><a href="/wiki/Gravel" title="Gravel">Gravel</a></li> <li><a href="/wiki/Peat" title="Peat">Peat</a></li> <li><a href="/wiki/Loam" title="Loam">Loam</a></li> <li><a href="/wiki/Loess" title="Loess">Loess</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:10.0em;font-weight: normal;">Properties</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hydraulic_conductivity" title="Hydraulic conductivity">Hydraulic conductivity</a></li> <li><a href="/wiki/Water_content" title="Water content">Water content</a></li> <li><a href="/wiki/Void_ratio" title="Void ratio">Void ratio</a></li> <li><a href="/wiki/Bulk_density" title="Bulk density">Bulk density</a></li> <li><a href="/wiki/Thixotropy" title="Thixotropy">Thixotropy</a></li> <li><a href="/wiki/Reynolds%27_dilatancy" class="mw-redirect" title="Reynolds' dilatancy">Reynolds' dilatancy</a></li> <li><a href="/wiki/Angle_of_repose" title="Angle of repose">Angle of repose</a></li> <li><a href="/wiki/Friction#Angle_of_friction" title="Friction">Friction angle</a></li> <li><a href="/wiki/Cohesion_(geology)" title="Cohesion (geology)">Cohesion</a></li> <li><a href="/wiki/Porosity" title="Porosity">Porosity</a></li> <li><a href="/wiki/Permeability_(earth_sciences)" class="mw-redirect" title="Permeability (earth sciences)">Permeability</a></li> <li><a href="/wiki/Specific_storage" title="Specific storage">Specific storage</a></li> <li><a href="/wiki/Shear_strength_(soil)" title="Shear strength (soil)">Shear strength</a></li> <li><a href="/wiki/Soil_liquefaction" title="Soil liquefaction">Sensitivity</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Structures <br />(<a href="/wiki/Soil-structure_interaction" title="Soil-structure interaction">Interaction</a>)</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:10.0em;font-weight: normal;">Natural features</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Topography" title="Topography">Topography</a></li> <li><a href="/wiki/Vegetation" title="Vegetation">Vegetation</a></li> <li><a href="/wiki/Terrain" title="Terrain">Terrain</a></li> <li><a href="/wiki/Topsoil" title="Topsoil">Topsoil</a></li> <li><a href="/wiki/Water_table" title="Water table">Water table</a></li> <li><a href="/wiki/Bedrock" title="Bedrock">Bedrock</a></li> <li><a href="/wiki/Subgrade" title="Subgrade">Subgrade</a></li> <li><a href="/wiki/Subsoil" title="Subsoil">Subsoil</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:10.0em;font-weight: normal;"><a href="/wiki/Earthworks_(engineering)" title="Earthworks (engineering)">Earthworks</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li>Shoring structures <ul><li><a href="/wiki/Retaining_wall" title="Retaining wall">Retaining walls</a></li> <li><a href="/wiki/Gabion" title="Gabion">Gabion</a></li> <li><a href="/wiki/Ground_freezing" title="Ground freezing">Ground freezing</a></li> <li><a href="/wiki/Mechanically_stabilized_earth" title="Mechanically stabilized earth">Mechanically stabilized earth</a></li> <li><a href="/wiki/Pressure_grouting" title="Pressure grouting">Pressure grouting</a></li> <li><a href="/wiki/Slurry_wall" title="Slurry wall">Slurry wall</a></li> <li><a href="/wiki/Soil_nailing" title="Soil nailing">Soil nailing</a></li> <li><a href="/wiki/Tieback_(geotechnical)" title="Tieback (geotechnical)">Tieback</a></li></ul></li> <li><a href="/wiki/Land_development" title="Land development">Land development</a></li> <li><a href="/wiki/Landfill" title="Landfill">Landfill</a></li> <li><a href="/wiki/Digging" title="Digging">Excavation</a></li> <li><a href="/wiki/Trench" title="Trench">Trench</a></li> <li><a href="/wiki/Embankment_(earthworks)" title="Embankment (earthworks)">Embankment</a></li> <li><a href="/wiki/Cut_(earthworks)" title="Cut (earthworks)">Cut</a></li> <li><a href="/wiki/Causeway" title="Causeway">Causeway</a></li> <li><a href="/wiki/Terrace_(earthworks)" title="Terrace (earthworks)">Terracing</a></li> <li><a href="/wiki/Tunnel#Cut-and-cover" title="Tunnel">Cut-and-cover</a></li> <li><a href="/wiki/Cut_and_fill" title="Cut and fill">Cut and fill</a></li> <li><a href="/wiki/Fill_dirt" title="Fill dirt">Fill dirt</a></li> <li><a href="/wiki/Grading_(engineering)" class="mw-redirect" title="Grading (engineering)">Grading</a></li> <li><a href="/wiki/Land_reclamation" title="Land reclamation">Land reclamation</a></li> <li><a href="/wiki/Track_bed" title="Track bed">Track bed</a></li> <li><a href="/wiki/Erosion_control" title="Erosion control">Erosion control</a></li> <li><a href="/wiki/Earth_structure" title="Earth structure">Earth structure</a></li> <li><a href="/wiki/Expanded_clay_aggregate" title="Expanded clay aggregate">Expanded clay aggregate</a></li> <li><a href="/wiki/Crushed_stone" title="Crushed stone">Crushed stone</a></li> <li><a href="/wiki/Geosynthetics" title="Geosynthetics">Geosynthetics</a> <ul><li><a href="/wiki/Geotextile" title="Geotextile">Geotextile</a></li> <li><a href="/wiki/Geomembrane" title="Geomembrane">Geomembrane</a></li> <li><a href="/wiki/Geosynthetic_clay_liner" title="Geosynthetic clay liner">Geosynthetic clay liner</a></li> <li><a href="/wiki/Cellular_confinement" title="Cellular confinement">Cellular confinement</a></li></ul></li> <li><a href="/wiki/Infiltration_(hydrology)" title="Infiltration (hydrology)">Infiltration</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:10.0em;font-weight: normal;"><a href="/wiki/Foundation_(engineering)" title="Foundation (engineering)">Foundations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Shallow_foundation" title="Shallow foundation">Shallow</a></li> <li><a href="/wiki/Deep_foundation" class="mw-redirect" title="Deep foundation">Deep</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Soil_mechanics" title="Soil mechanics">Mechanics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:10.0em;font-weight: normal;">Forces</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a 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title="Road">Slab stabilisation</a></li></ul></li> <li><a href="/wiki/Bearing_capacity" title="Bearing capacity">Bearing capacity</a> * <a href="/wiki/Stress_distribution_in_soil" title="Stress distribution in soil">Stress distribution in soil</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Software" title="Software">Numerical analysis <br />software</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/SEEP2D" title="SEEP2D">SEEP2D</a></li> <li><a href="/w/index.php?title=STABL&action=edit&redlink=1" class="new" title="STABL (page does not exist)">STABL</a></li> <li><a href="/wiki/SVFlux" title="SVFlux">SVFlux</a></li> <li><a href="/wiki/SVSlope" title="SVSlope">SVSlope</a></li> <li><a href="/wiki/UTEXAS" title="UTEXAS">UTEXAS</a></li> <li><a href="/wiki/Plaxis" 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