List of second moments of area

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</div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">List of second moments of area</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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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"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=List_of_area_moments_of_inertia&redirect=no" class="mw-redirect" title="List of area moments of inertia">List of area moments of inertia</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p class="mw-empty-elt"> </p><p>The following is a <b>list of second moments of area</b> of some shapes. The <a href="/wiki/Second_moment_of_area" title="Second moment of area">second moment of area</a>, also known as area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with respect to an arbitrary axis. The <a href="/wiki/Physical_unit" class="mw-redirect" title="Physical unit">unit</a> of dimension of the second moment of area is length to fourth power, <a href="/wiki/Unit_of_length" title="Unit of length">L</a><sup>4</sup>, and should not be confused with the <a href="/wiki/Mass_moment_of_inertia" class="mw-redirect" title="Mass moment of inertia">mass moment of inertia</a>. If the piece is thin, however, the mass moment of inertia equals the <a href="/wiki/Area_density" title="Area density">area density</a> times the area moment of inertia. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Second_moments_of_area">Second moments of area</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_second_moments_of_area&action=edit&section=1" title="Edit section: Second moments of area"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Please note that for the second moment of area equations in the below table: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{x}=\iint _{A}y^{2}\,dx\,dy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{x}=\iint _{A}y^{2}\,dx\,dy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2579f37f3984934c0383073453cc9a7b03a98360" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.571ex; height:5.676ex;" alt="{\displaystyle I_{x}=\iint _{A}y^{2}\,dx\,dy}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{y}=\iint _{A}x^{2}\,dx\,dy.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{y}=\iint _{A}x^{2}\,dx\,dy.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/310c8db0f7292b92d3ff065573873fd7d30826f3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.264ex; height:5.676ex;" alt="{\displaystyle I_{y}=\iint _{A}x^{2}\,dx\,dy.}"></span> </p> <table class="wikitable"> <tbody><tr> <th>Description </th> <th>Figure </th> <th>Second moment of area </th> <th>Comment </th></tr> <tr> <td>A filled circular area of radius <i>r</i></td> <td><span typeof="mw:File"><a href="/wiki/File:Moment_of_area_of_a_circle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Moment_of_area_of_a_circle.svg/230px-Moment_of_area_of_a_circle.svg.png" decoding="async" width="230" height="230" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Moment_of_area_of_a_circle.svg/345px-Moment_of_area_of_a_circle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Moment_of_area_of_a_circle.svg/460px-Moment_of_area_of_a_circle.svg.png 2x" data-file-width="341" data-file-height="341" /></a></span> </td> <td><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 {\begin{aligned}I_{x}&={\frac {\pi }{4}}r^{4}\\[3pt]I_{y}&={\frac {\pi }{4}}r^{4}\\[3pt]I_{z}&={\frac {\pi }{2}}r^{4}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.6em 0.6em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I_{x}&={\frac {\pi }{4}}r^{4}\\[3pt]I_{y}&={\frac {\pi }{4}}r^{4}\\[3pt]I_{z}&={\frac {\pi }{2}}r^{4}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/278e00f090677da6ac3cd226e78a98bf21e3e8ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.095ex; margin-bottom: -0.243ex; width:10.317ex; height:15.843ex;" alt="{\displaystyle {\begin{aligned}I_{x}&={\frac {\pi }{4}}r^{4}\\[3pt]I_{y}&={\frac {\pi }{4}}r^{4}\\[3pt]I_{z}&={\frac {\pi }{2}}r^{4}\end{aligned}}}"></span> <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></td> <td><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 I_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f8f43aaaf2e7ed525e6d3b690f2f7d31cca1478" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.025ex; height:2.509ex;" alt="{\displaystyle I_{z}}"></span> is the <a href="/wiki/Second_polar_moment_of_area" title="Second polar moment of area">second polar moment of area</a>. </td></tr> <tr> <td>An <a href="/wiki/Annulus_(mathematics)" title="Annulus (mathematics)">annulus</a> of inner radius <i>r</i><sub>1</sub> and outer radius <i>r</i><sub>2</sub></td> <td><span typeof="mw:File"><a href="/wiki/File:Moment_of_area_of_an_annulus.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Moment_of_area_of_an_annulus.svg/230px-Moment_of_area_of_an_annulus.svg.png" decoding="async" width="230" height="230" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Moment_of_area_of_an_annulus.svg/345px-Moment_of_area_of_an_annulus.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Moment_of_area_of_an_annulus.svg/460px-Moment_of_area_of_an_annulus.svg.png 2x" data-file-width="341" data-file-height="341" /></a></span> </td> <td><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 {\begin{aligned}I_{x}&={\frac {\pi }{4}}\left({r_{2}}^{4}-{r_{1}}^{4}\right)\\[3pt]I_{y}&={\frac {\pi }{4}}\left({r_{2}}^{4}-{r_{1}}^{4}\right)\\[3pt]I_{z}&={\frac {\pi }{2}}\left({r_{2}}^{4}-{r_{1}}^{4}\right)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.6em 0.6em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I_{x}&={\frac {\pi }{4}}\left({r_{2}}^{4}-{r_{1}}^{4}\right)\\[3pt]I_{y}&={\frac {\pi }{4}}\left({r_{2}}^{4}-{r_{1}}^{4}\right)\\[3pt]I_{z}&={\frac {\pi }{2}}\left({r_{2}}^{4}-{r_{1}}^{4}\right)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2c0dc505866915f275ecd4962638f82248bf853" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.095ex; margin-bottom: -0.243ex; width:19.885ex; height:15.843ex;" alt="{\displaystyle {\begin{aligned}I_{x}&={\frac {\pi }{4}}\left({r_{2}}^{4}-{r_{1}}^{4}\right)\\[3pt]I_{y}&={\frac {\pi }{4}}\left({r_{2}}^{4}-{r_{1}}^{4}\right)\\[3pt]I_{z}&={\frac {\pi }{2}}\left({r_{2}}^{4}-{r_{1}}^{4}\right)\end{aligned}}}"></span> </td> <td> <p>For thin tubes, <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 r\equiv r_{1}\approx r_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>≡</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≈</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\equiv r_{1}\approx r_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62838ddeab34ac038857880048d4430e99008831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.451ex; height:2.009ex;" alt="{\displaystyle r\equiv r_{1}\approx r_{2}}"></span> and <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 r_{2}\equiv r_{1}+t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≡</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}\equiv r_{1}+t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c9b35fff855299bd178468a7baebb9cfce886e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.984ex; height:2.343ex;" alt="{\displaystyle r_{2}\equiv r_{1}+t}"></span> and so to first order in <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 t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>, <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 r_{2}^{4}=r_{1}^{4}+4r_{1}^{3}t+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo>+</mo> <mn>4</mn> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mi>t</mi> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}^{4}=r_{1}^{4}+4r_{1}^{3}t+\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01433968a54455f5537a0ac81cd17b02fb3bed83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.813ex; height:3.176ex;" alt="{\displaystyle r_{2}^{4}=r_{1}^{4}+4r_{1}^{3}t+\cdots }"></span>. So, for a thin tube, <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 I_{x}=I_{y}\approx \pi r^{3}t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>≈</mo> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{x}=I_{y}\approx \pi r^{3}t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2ae1aeeaa35cb7b4646e1405ee1f435b412ab3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.739ex; height:3.343ex;" alt="{\displaystyle I_{x}=I_{y}\approx \pi r^{3}t}"></span> and <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 I_{z}\approx 2\pi r^{3}t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>≈</mo> <mn>2</mn> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{z}\approx 2\pi r^{3}t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bbb36800140bdbf0ea79f8d2361d032a9e19359" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.56ex; height:3.009ex;" alt="{\displaystyle I_{z}\approx 2\pi r^{3}t}"></span>. </p><p><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 I_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f8f43aaaf2e7ed525e6d3b690f2f7d31cca1478" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.025ex; height:2.509ex;" alt="{\displaystyle I_{z}}"></span> is the <a href="/wiki/Second_polar_moment_of_area" title="Second polar moment of area">second polar moment of area</a>. </p> </td></tr> <tr> <td>A filled <a href="/wiki/Circular_sector" title="Circular sector">circular sector</a> of angle <i>θ</i> in <a href="/wiki/Radian" title="Radian">radians</a> and radius <i>r</i> with respect to an axis through the centroid of the sector and the center of the circle</td> <td><span typeof="mw:File"><a href="/wiki/File:Moment_of_area_of_a_circular_sector.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Moment_of_area_of_a_circular_sector.svg/230px-Moment_of_area_of_a_circular_sector.svg.png" decoding="async" width="230" height="230" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Moment_of_area_of_a_circular_sector.svg/345px-Moment_of_area_of_a_circular_sector.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Moment_of_area_of_a_circular_sector.svg/460px-Moment_of_area_of_a_circular_sector.svg.png 2x" data-file-width="341" data-file-height="341" /></a></span></td> <td><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 I_{x}=\left(\theta -\sin \theta \right){\frac {r^{4}}{8}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>θ</mi> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mn>8</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{x}=\left(\theta -\sin \theta \right){\frac {r^{4}}{8}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/357c9d8a735eec76b3b4aa757870297a39ab76d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.693ex; height:5.676ex;" alt="{\displaystyle I_{x}=\left(\theta -\sin \theta \right){\frac {r^{4}}{8}}}"></span></td> <td>This formula is valid only for 0 ≤ <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="{\textstyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a11744bd71a5eb6efe4f28e12ca57f874d82658c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\textstyle \theta }"></span> ≤ <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="{\textstyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/584b74e57567454e6a48e040ec231214be5b8c54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\textstyle 2\pi }"></span> </td></tr> <tr> <td>A filled semicircle with radius <i>r</i> with respect to a horizontal line passing through the centroid of the area</td> <td><span typeof="mw:File"><a href="/wiki/File:Moment_of_area_of_a_semicircle_through_the_centroid.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Moment_of_area_of_a_semicircle_through_the_centroid.svg/230px-Moment_of_area_of_a_semicircle_through_the_centroid.svg.png" decoding="async" width="230" height="230" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Moment_of_area_of_a_semicircle_through_the_centroid.svg/345px-Moment_of_area_of_a_semicircle_through_the_centroid.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/50/Moment_of_area_of_a_semicircle_through_the_centroid.svg/460px-Moment_of_area_of_a_semicircle_through_the_centroid.svg.png 2x" data-file-width="341" data-file-height="341" /></a></span> </td> <td><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 {\begin{aligned}I_{x}&=\left({\frac {\pi }{8}}-{\frac {8}{9\pi }}\right)r^{4}\approx 0.1098r^{4}\\[3pt]I_{y}&={\frac {\pi r^{4}}{8}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.6em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>8</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mrow> <mn>9</mn> <mi>π</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>≈</mo> <mn>0.1098</mn> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mn>8</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I_{x}&=\left({\frac {\pi }{8}}-{\frac {8}{9\pi }}\right)r^{4}\approx 0.1098r^{4}\\[3pt]I_{y}&={\frac {\pi r^{4}}{8}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddc0a8acc04324651c3bc2579b1cb8452e1ef66d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.492ex; margin-bottom: -0.179ex; width:31.956ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}I_{x}&=\left({\frac {\pi }{8}}-{\frac {8}{9\pi }}\right)r^{4}\approx 0.1098r^{4}\\[3pt]I_{y}&={\frac {\pi r^{4}}{8}}\end{aligned}}}"></span> <sup id="cite_ref-semicircle_2-0" class="reference"><a href="#cite_note-semicircle-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></td> <td> </td></tr> <tr> <td>A filled semicircle as above but with respect to an axis collinear with the base</td> <td><span typeof="mw:File"><a href="/wiki/File:Moment_of_area_of_a_semicircle_through_the_base.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Moment_of_area_of_a_semicircle_through_the_base.svg/230px-Moment_of_area_of_a_semicircle_through_the_base.svg.png" decoding="async" width="230" height="230" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Moment_of_area_of_a_semicircle_through_the_base.svg/345px-Moment_of_area_of_a_semicircle_through_the_base.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Moment_of_area_of_a_semicircle_through_the_base.svg/460px-Moment_of_area_of_a_semicircle_through_the_base.svg.png 2x" data-file-width="341" data-file-height="341" /></a></span> </td> <td><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 {\begin{aligned}I_{x}&={\frac {\pi r^{4}}{8}}\\[3pt]I_{y}&={\frac {\pi r^{4}}{8}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.6em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mn>8</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mn>8</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I_{x}&={\frac {\pi r^{4}}{8}}\\[3pt]I_{y}&={\frac {\pi r^{4}}{8}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/277b9ec1de780c3a576a1007c4d2458360987062" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:10.317ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}I_{x}&={\frac {\pi r^{4}}{8}}\\[3pt]I_{y}&={\frac {\pi r^{4}}{8}}\end{aligned}}}"></span> <sup id="cite_ref-semicircle_2-1" class="reference"><a href="#cite_note-semicircle-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></td> <td><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 I_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eed0dfcd8b1ce6eb2e6b7fbc426b895507d53004" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.196ex; height:2.509ex;" alt="{\displaystyle I_{x}}"></span>: This is a consequence of the <a href="/wiki/Parallel_axis_theorem" title="Parallel axis theorem">parallel axis theorem</a> and the fact that the distance between the x axes of the previous one and this one 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="{\textstyle {\frac {4r}{3\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>r</mi> </mrow> <mrow> <mn>3</mn> <mi>π</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {4r}{3\pi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0414c234858ee106833176616e040ab36257979" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.6ex; height:3.676ex;" alt="{\textstyle {\frac {4r}{3\pi }}}"></span> </td></tr> <tr> <td>A filled quarter circle with radius <i>r</i> with the axes passing through the bases</td> <td><span typeof="mw:File"><a href="/wiki/File:Moment_of_area_of_a_quarter_circle_through_the_base.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Moment_of_area_of_a_quarter_circle_through_the_base.svg/230px-Moment_of_area_of_a_quarter_circle_through_the_base.svg.png" decoding="async" width="230" height="230" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Moment_of_area_of_a_quarter_circle_through_the_base.svg/345px-Moment_of_area_of_a_quarter_circle_through_the_base.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/50/Moment_of_area_of_a_quarter_circle_through_the_base.svg/460px-Moment_of_area_of_a_quarter_circle_through_the_base.svg.png 2x" data-file-width="341" data-file-height="341" /></a></span> </td> <td><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 {\begin{aligned}I_{x}&={\frac {\pi r^{4}}{16}}\\[3pt]I_{y}&={\frac {\pi r^{4}}{16}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.6em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mn>16</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mn>16</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I_{x}&={\frac {\pi r^{4}}{16}}\\[3pt]I_{y}&={\frac {\pi r^{4}}{16}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/764c8723a3141ac5d7ad49ca7abd6d69ef13984b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:10.317ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}I_{x}&={\frac {\pi r^{4}}{16}}\\[3pt]I_{y}&={\frac {\pi r^{4}}{16}}\end{aligned}}}"></span> <sup id="cite_ref-quartercircle_3-0" class="reference"><a href="#cite_note-quartercircle-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></td> <td> </td></tr> <tr> <td>A filled quarter circle with radius <i>r</i> with the axes passing through the centroid</td> <td><span typeof="mw:File"><a href="/wiki/File:Moment_of_area_of_a_quarter_circle_through_the_centroid.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Moment_of_area_of_a_quarter_circle_through_the_centroid.svg/230px-Moment_of_area_of_a_quarter_circle_through_the_centroid.svg.png" decoding="async" width="230" height="230" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Moment_of_area_of_a_quarter_circle_through_the_centroid.svg/345px-Moment_of_area_of_a_quarter_circle_through_the_centroid.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Moment_of_area_of_a_quarter_circle_through_the_centroid.svg/460px-Moment_of_area_of_a_quarter_circle_through_the_centroid.svg.png 2x" data-file-width="341" data-file-height="341" /></a></span> </td> <td><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 {\begin{aligned}I_{x}&=\left({\frac {\pi }{16}}-{\frac {4}{9\pi }}\right)r^{4}\approx 0.0549r^{4}\\[3pt]I_{y}&=\left({\frac {\pi }{16}}-{\frac {4}{9\pi }}\right)r^{4}\approx 0.0549r^{4}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.6em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>16</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mrow> <mn>9</mn> <mi>π</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>≈</mo> <mn>0.0549</mn> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>16</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mrow> <mn>9</mn> <mi>π</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>≈</mo> <mn>0.0549</mn> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I_{x}&=\left({\frac {\pi }{16}}-{\frac {4}{9\pi }}\right)r^{4}\approx 0.0549r^{4}\\[3pt]I_{y}&=\left({\frac {\pi }{16}}-{\frac {4}{9\pi }}\right)r^{4}\approx 0.0549r^{4}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa80a92550ec7bef1d6b8d5acc4ed7a04ec84bfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:32.949ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}I_{x}&=\left({\frac {\pi }{16}}-{\frac {4}{9\pi }}\right)r^{4}\approx 0.0549r^{4}\\[3pt]I_{y}&=\left({\frac {\pi }{16}}-{\frac {4}{9\pi }}\right)r^{4}\approx 0.0549r^{4}\end{aligned}}}"></span> <sup id="cite_ref-quartercircle_3-1" class="reference"><a href="#cite_note-quartercircle-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></td> <td>This is a consequence of the <a href="/wiki/Parallel_axis_theorem" title="Parallel axis theorem">parallel axis theorem</a> and the fact that the distance between these two axes 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="{\textstyle {\frac {4r}{3\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>r</mi> </mrow> <mrow> <mn>3</mn> <mi>π</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {4r}{3\pi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0414c234858ee106833176616e040ab36257979" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.6ex; height:3.676ex;" alt="{\textstyle {\frac {4r}{3\pi }}}"></span> </td></tr> <tr> <td>A filled <a href="/wiki/Ellipse" title="Ellipse">ellipse</a> whose radius along the <i>x</i>-axis is <i>a</i> and whose radius along the <i>y</i>-axis is <i>b</i></td> <td><span typeof="mw:File"><a href="/wiki/File:Moment_of_area_of_an_ellipse.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Moment_of_area_of_an_ellipse.svg/230px-Moment_of_area_of_an_ellipse.svg.png" decoding="async" width="230" height="230" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Moment_of_area_of_an_ellipse.svg/345px-Moment_of_area_of_an_ellipse.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Moment_of_area_of_an_ellipse.svg/460px-Moment_of_area_of_an_ellipse.svg.png 2x" data-file-width="341" data-file-height="341" /></a></span> </td> <td><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 {\begin{aligned}I_{x}&={\frac {\pi }{4}}ab^{3}\\[3pt]I_{y}&={\frac {\pi }{4}}a^{3}b\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.6em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mi>a</mi> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>b</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I_{x}&={\frac {\pi }{4}}ab^{3}\\[3pt]I_{y}&={\frac {\pi }{4}}a^{3}b\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e07bb49e05192b23a5e658bfb015da92b15cb4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.313ex; margin-bottom: -0.192ex; width:11.495ex; height:10.176ex;" alt="{\displaystyle {\begin{aligned}I_{x}&={\frac {\pi }{4}}ab^{3}\\[3pt]I_{y}&={\frac {\pi }{4}}a^{3}b\end{aligned}}}"></span></td> <td> </td></tr> <tr> <td>A filled rectangular area with a base width of <i>b</i> and height <i>h</i></td> <td><span typeof="mw:File"><a href="/wiki/File:Moment_of_area_of_a_rectangle_through_the_centroid.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Moment_of_area_of_a_rectangle_through_the_centroid.svg/230px-Moment_of_area_of_a_rectangle_through_the_centroid.svg.png" decoding="async" width="230" height="230" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Moment_of_area_of_a_rectangle_through_the_centroid.svg/345px-Moment_of_area_of_a_rectangle_through_the_centroid.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/82/Moment_of_area_of_a_rectangle_through_the_centroid.svg/460px-Moment_of_area_of_a_rectangle_through_the_centroid.svg.png 2x" data-file-width="341" data-file-height="341" /></a></span> </td> <td><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 {\begin{aligned}I_{x}&={\frac {bh^{3}}{12}}\\[3pt]I_{y}&={\frac {b^{3}h}{12}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.6em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mn>12</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>h</mi> </mrow> <mn>12</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I_{x}&={\frac {bh^{3}}{12}}\\[3pt]I_{y}&={\frac {b^{3}h}{12}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21d0bddcf815ee4746396673832c2d9458d131f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:10.272ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}I_{x}&={\frac {bh^{3}}{12}}\\[3pt]I_{y}&={\frac {b^{3}h}{12}}\end{aligned}}}"></span> <sup id="cite_ref-rect_4-0" class="reference"><a href="#cite_note-rect-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></td> <td> </td></tr> <tr> <td>A filled rectangular area as above but with respect to an axis collinear with the base</td> <td><span typeof="mw:File"><a href="/wiki/File:Moment_of_area_of_a_rectangle_through_the_base.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Moment_of_area_of_a_rectangle_through_the_base.svg/230px-Moment_of_area_of_a_rectangle_through_the_base.svg.png" decoding="async" width="230" height="230" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Moment_of_area_of_a_rectangle_through_the_base.svg/345px-Moment_of_area_of_a_rectangle_through_the_base.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Moment_of_area_of_a_rectangle_through_the_base.svg/460px-Moment_of_area_of_a_rectangle_through_the_base.svg.png 2x" data-file-width="341" data-file-height="341" /></a></span> </td> <td><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 {\begin{aligned}I_{x}&={\frac {bh^{3}}{3}}\\[3pt]I_{y}&={\frac {b^{3}h}{3}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.6em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mn>3</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>h</mi> </mrow> <mn>3</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I_{x}&={\frac {bh^{3}}{3}}\\[3pt]I_{y}&={\frac {b^{3}h}{3}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9900f74a013e4b660eebfcf99ea54f3e2a42320a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:10.272ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}I_{x}&={\frac {bh^{3}}{3}}\\[3pt]I_{y}&={\frac {b^{3}h}{3}}\end{aligned}}}"></span> <sup id="cite_ref-rect_4-1" class="reference"><a href="#cite_note-rect-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></td> <td>This is a result from the <a href="/wiki/Parallel_axis_theorem" title="Parallel axis theorem">parallel axis theorem</a> </td></tr> <tr> <td>A hollow <a href="/wiki/Rectangle" title="Rectangle">rectangle</a> with an inner rectangle whose width is <i>b</i><sub>1</sub> and whose height is <i>h</i><sub>1</sub></td> <td><span typeof="mw:File"><a href="/wiki/File:Moment_of_area_of_a_hollow_rectangle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Moment_of_area_of_a_hollow_rectangle.svg/230px-Moment_of_area_of_a_hollow_rectangle.svg.png" decoding="async" width="230" height="230" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Moment_of_area_of_a_hollow_rectangle.svg/345px-Moment_of_area_of_a_hollow_rectangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Moment_of_area_of_a_hollow_rectangle.svg/460px-Moment_of_area_of_a_hollow_rectangle.svg.png 2x" data-file-width="341" data-file-height="341" /></a></span> </td> <td><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 {\begin{aligned}I_{x}&={\frac {bh^{3}-b_{1}{h_{1}}^{3}}{12}}\\[3pt]I_{y}&={\frac {b^{3}h-{b_{1}}^{3}h_{1}}{12}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.6em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mn>12</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>h</mi> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mn>12</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I_{x}&={\frac {bh^{3}-b_{1}{h_{1}}^{3}}{12}}\\[3pt]I_{y}&={\frac {b^{3}h-{b_{1}}^{3}h_{1}}{12}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e3784d81914f238868aec3ea913cc280a599e7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:18.612ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}I_{x}&={\frac {bh^{3}-b_{1}{h_{1}}^{3}}{12}}\\[3pt]I_{y}&={\frac {b^{3}h-{b_{1}}^{3}h_{1}}{12}}\end{aligned}}}"></span></td> <td> </td></tr> <tr> <td>A filled triangular area with a base width of <i>b</i>, height <i>h</i> and top vertex displacement <i>a</i>, with respect to an axis through the centroid</td> <td><figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/File:Moment_of_inertia_on_a_triangle_through_centroide_with_dimension_%27a%27.svg" class="mw-file-description" title="The figure presents a triangle with dimensions 'b', 'h' and 'a', along with axes 'x' and 'y' that pass through the centroid."><img alt="The figure presents a triangle with dimensions 'b', 'h' and 'a', along with axes 'x' and 'y' that pass through the centroid." src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Moment_of_inertia_on_a_triangle_through_centroide_with_dimension_%27a%27.svg/230px-Moment_of_inertia_on_a_triangle_through_centroide_with_dimension_%27a%27.svg.png" decoding="async" width="230" height="230" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Moment_of_inertia_on_a_triangle_through_centroide_with_dimension_%27a%27.svg/345px-Moment_of_inertia_on_a_triangle_through_centroide_with_dimension_%27a%27.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Moment_of_inertia_on_a_triangle_through_centroide_with_dimension_%27a%27.svg/460px-Moment_of_inertia_on_a_triangle_through_centroide_with_dimension_%27a%27.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>The figure presents a triangle with dimensions 'b', 'h' and 'a', along with axes 'x' and 'y' that pass through the centroid.</figcaption></figure> </td> <td><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 {\begin{aligned}I_{x}&={\frac {bh^{3}}{36}}\\[3pt]I_{y}&={\frac {b^{3}h-b^{2}ha+bha^{2}}{36}}\\[3pt]I_{xy}&=-{\frac {bh^{2}}{72}}(b-2a)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.6em 0.6em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mn>36</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>h</mi> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>h</mi> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>h</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>36</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>72</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mn>2</mn> <mi>a</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I_{x}&={\frac {bh^{3}}{36}}\\[3pt]I_{y}&={\frac {b^{3}h-b^{2}ha+bha^{2}}{36}}\\[3pt]I_{xy}&=-{\frac {bh^{2}}{72}}(b-2a)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/779393bd3a378cb18b44f551ad3e88c325584e00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.838ex; width:26.011ex; height:18.843ex;" alt="{\displaystyle {\begin{aligned}I_{x}&={\frac {bh^{3}}{36}}\\[3pt]I_{y}&={\frac {b^{3}h-b^{2}ha+bha^{2}}{36}}\\[3pt]I_{xy}&=-{\frac {bh^{2}}{72}}(b-2a)\end{aligned}}}"></span> <sup id="cite_ref-tri_5-0" class="reference"><a href="#cite_note-tri-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td>A filled triangular area as above but with respect to an axis collinear with the base</td> <td><figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/File:Moment_of_inertia_on_a_traingle_through_the_base_with_dimension_%27a%27.svg" class="mw-file-description" title="The figure presents a triangle with dimensions 'b', 'h' and 'a', along with axes 'x' and 'y', 'x' being collinear with the base."><img alt="The figure presents a triangle with dimensions 'b', 'h' and 'a', along with axes 'x' and 'y', 'x' being collinear with the base." src="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/Moment_of_inertia_on_a_traingle_through_the_base_with_dimension_%27a%27.svg/230px-Moment_of_inertia_on_a_traingle_through_the_base_with_dimension_%27a%27.svg.png" decoding="async" width="230" height="230" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/Moment_of_inertia_on_a_traingle_through_the_base_with_dimension_%27a%27.svg/345px-Moment_of_inertia_on_a_traingle_through_the_base_with_dimension_%27a%27.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/92/Moment_of_inertia_on_a_traingle_through_the_base_with_dimension_%27a%27.svg/460px-Moment_of_inertia_on_a_traingle_through_the_base_with_dimension_%27a%27.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>The figure presents a triangle with dimensions 'b', 'h' and 'a', along with axes 'x' and 'y', 'x' being collinear with the base.</figcaption></figure> </td> <td><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 {\begin{aligned}I_{x}&={\frac {bh^{3}}{12}}\\[3pt]I_{y}&={\frac {b^{3}h+b^{2}ha+bha^{2}}{12}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.6em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mn>12</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>h</mi> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>h</mi> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>h</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>12</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I_{x}&={\frac {bh^{3}}{12}}\\[3pt]I_{y}&={\frac {b^{3}h+b^{2}ha+bha^{2}}{12}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bc71985a387580b8666b40ce5cdcb6ad7869cd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:25.194ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}I_{x}&={\frac {bh^{3}}{12}}\\[3pt]I_{y}&={\frac {b^{3}h+b^{2}ha+bha^{2}}{12}}\end{aligned}}}"></span> <sup id="cite_ref-tri_5-1" class="reference"><a href="#cite_note-tri-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></td> <td>This is a consequence of the <a href="/wiki/Parallel_axis_theorem" title="Parallel axis theorem">parallel axis theorem</a> </td></tr> <tr> <td>An equal legged angle, commonly found in engineering applications</td> <td><span typeof="mw:File"><a href="/wiki/File:Second_Moment_of_Area_Angle.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Second_Moment_of_Area_Angle.jpg/230px-Second_Moment_of_Area_Angle.jpg" decoding="async" width="230" height="208" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Second_Moment_of_Area_Angle.jpg/345px-Second_Moment_of_Area_Angle.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Second_Moment_of_Area_Angle.jpg/460px-Second_Moment_of_Area_Angle.jpg 2x" data-file-width="872" data-file-height="788" /></a></span> </td> <td><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 {\begin{aligned}I_{x}=I_{y}&={\frac {t(5L^{2}-5Lt+t^{2})(L^{2}-Lt+t^{2})}{12(2L-t)}}\\[3pt]I_{(xy)}&={\frac {L^{2}t(L-t)^{2}}{4(t-2L)}}\\[3pt]I_{a}&={\frac {t(2L-t)(2L^{2}-2Lt+t^{2})}{12}}\\[3pt]I_{b}&={\frac {t(2L^{4}-4L^{3}t+8L^{2}t^{2}-6Lt^{3}+t^{4})}{12(2L-t)}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.6em 0.6em 0.6em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>t</mi> <mo stretchy="false">(</mo> <mn>5</mn> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>5</mn> <mi>L</mi> <mi>t</mi> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>L</mi> <mi>t</mi> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>12</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mi>L</mi> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>t</mi> <mo stretchy="false">(</mo> <mi>L</mi> <mo>−</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mn>2</mn> <mi>L</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>t</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>L</mi> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>L</mi> <mi>t</mi> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mn>12</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>t</mi> <mo stretchy="false">(</mo> <mn>2</mn> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>t</mi> <mo>+</mo> <mn>8</mn> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>6</mn> <mi>L</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>12</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mi>L</mi> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I_{x}=I_{y}&={\frac {t(5L^{2}-5Lt+t^{2})(L^{2}-Lt+t^{2})}{12(2L-t)}}\\[3pt]I_{(xy)}&={\frac {L^{2}t(L-t)^{2}}{4(t-2L)}}\\[3pt]I_{a}&={\frac {t(2L-t)(2L^{2}-2Lt+t^{2})}{12}}\\[3pt]I_{b}&={\frac {t(2L^{4}-4L^{3}t+8L^{2}t^{2}-6Lt^{3}+t^{4})}{12(2L-t)}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a62f26c9790cae3a20588fa574b3d34f268216ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.505ex; width:46.728ex; height:28.176ex;" alt="{\displaystyle {\begin{aligned}I_{x}=I_{y}&={\frac {t(5L^{2}-5Lt+t^{2})(L^{2}-Lt+t^{2})}{12(2L-t)}}\\[3pt]I_{(xy)}&={\frac {L^{2}t(L-t)^{2}}{4(t-2L)}}\\[3pt]I_{a}&={\frac {t(2L-t)(2L^{2}-2Lt+t^{2})}{12}}\\[3pt]I_{b}&={\frac {t(2L^{4}-4L^{3}t+8L^{2}t^{2}-6Lt^{3}+t^{4})}{12(2L-t)}}\end{aligned}}}"></span></td> <td><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 I_{(xy)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{(xy)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab469e0a61009ba86a768046fe1e1158f2af5d5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.292ex; height:3.009ex;" alt="{\displaystyle I_{(xy)}}"></span> is the often unused "product second moment of area", used to define principal axes </td></tr></tbody></table> <table class="wikitable mw-collapsible"> <tbody><tr> <th colspan="4" style="background: silver"><b><a href="/wiki/Regular_polygons" class="mw-redirect" title="Regular polygons">Regular polygons</a></b> </th></tr> <tr> <th>Description </th> <th>Figure </th> <th>Second moment of area </th> <th>Comment </th></tr> <tr> <td>A filled <a href="/wiki/Regular_triangle" class="mw-redirect" title="Regular triangle">regular</a> <a href="/wiki/Equilateral_triangle" title="Equilateral triangle">(equiliteral) triangle</a> with a side length of <i>a</i></td> <td><span typeof="mw:File"><a href="/wiki/File:Moment_of_area_of_a_regular_triangle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Moment_of_area_of_a_regular_triangle.svg/230px-Moment_of_area_of_a_regular_triangle.svg.png" decoding="async" width="230" height="230" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Moment_of_area_of_a_regular_triangle.svg/345px-Moment_of_area_of_a_regular_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/68/Moment_of_area_of_a_regular_triangle.svg/460px-Moment_of_area_of_a_regular_triangle.svg.png 2x" data-file-width="341" data-file-height="341" /></a></span> </td> <td><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 {\begin{aligned}I_{x}&={\frac {a^{4}}{32{\sqrt {3}}}}\approx 0.01804a^{4}\\[3pt]I_{y}&={\frac {a^{4}}{32{\sqrt {3}}}}\approx 0.01804a^{4}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.6em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow> <mn>32</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>≈</mo> <mn>0.01804</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow> <mn>32</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>≈</mo> <mn>0.01804</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I_{x}&={\frac {a^{4}}{32{\sqrt {3}}}}\approx 0.01804a^{4}\\[3pt]I_{y}&={\frac {a^{4}}{32{\sqrt {3}}}}\approx 0.01804a^{4}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04f1cdf57241346ae708bb33fa7f172beea6b06e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:25.309ex; height:13.843ex;" alt="{\displaystyle {\begin{aligned}I_{x}&={\frac {a^{4}}{32{\sqrt {3}}}}\approx 0.01804a^{4}\\[3pt]I_{y}&={\frac {a^{4}}{32{\sqrt {3}}}}\approx 0.01804a^{4}\end{aligned}}}"></span> <sup id="cite_ref-roark_6-0" class="reference"><a href="#cite_note-roark-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></td> <td>The result is valid for both a horizontal and a vertical axis through the centroid, and therefore is also valid for an axis with arbitrary direction that passes through the origin. <p>This holds true for all <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygons</a>. </p> </td></tr> <tr> <td>A filled <a href="/wiki/Square" title="Square">square</a> with a side length of <i>a</i></td> <td><span typeof="mw:File"><a href="/wiki/File:Moment_of_area_of_a_regular_square.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Moment_of_area_of_a_regular_square.svg/230px-Moment_of_area_of_a_regular_square.svg.png" decoding="async" width="230" height="230" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Moment_of_area_of_a_regular_square.svg/345px-Moment_of_area_of_a_regular_square.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Moment_of_area_of_a_regular_square.svg/460px-Moment_of_area_of_a_regular_square.svg.png 2x" data-file-width="341" data-file-height="341" /></a></span> </td> <td><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 {\begin{aligned}I_{x}&={\frac {a^{4}}{12}}\\[3pt]I_{y}&={\frac {a^{4}}{12}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.6em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mn>12</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mn>12</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I_{x}&={\frac {a^{4}}{12}}\\[3pt]I_{y}&={\frac {a^{4}}{12}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b6c0c9bede7b5176e57a93204ce52d9bcdea033" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:9.207ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}I_{x}&={\frac {a^{4}}{12}}\\[3pt]I_{y}&={\frac {a^{4}}{12}}\end{aligned}}}"></span><sup id="cite_ref-roark_6-1" class="reference"><a href="#cite_note-roark-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></td> <td>The result is valid for both a horizontal and a vertical axis through the centroid, and therefore is also valid for an axis with arbitrary direction that passes through the origin. <p>This holds true for all <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygons</a>. </p> </td></tr> <tr> <td>A filled <a href="/wiki/Regular_hexagon" class="mw-redirect" title="Regular hexagon">regular hexagon</a> with a side length of <i>a</i></td> <td><span typeof="mw:File"><a href="/wiki/File:Moment_of_area_of_a_regular_hexagon.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Moment_of_area_of_a_regular_hexagon.svg/230px-Moment_of_area_of_a_regular_hexagon.svg.png" decoding="async" width="230" height="230" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Moment_of_area_of_a_regular_hexagon.svg/345px-Moment_of_area_of_a_regular_hexagon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Moment_of_area_of_a_regular_hexagon.svg/460px-Moment_of_area_of_a_regular_hexagon.svg.png 2x" data-file-width="341" data-file-height="341" /></a></span> </td> <td><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 {\begin{aligned}I_{x}&={\frac {5{\sqrt {3}}}{16}}a^{4}\approx 0.54126a^{4}\\[3pt]I_{y}&={\frac {5{\sqrt {3}}}{16}}a^{4}\approx 0.54126a^{4}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.6em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> <mn>16</mn> </mfrac> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>≈</mo> <mn>0.54126</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> <mn>16</mn> </mfrac> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>≈</mo> <mn>0.54126</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I_{x}&={\frac {5{\sqrt {3}}}{16}}a^{4}\approx 0.54126a^{4}\\[3pt]I_{y}&={\frac {5{\sqrt {3}}}{16}}a^{4}\approx 0.54126a^{4}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/accebf245f55a7b16ced477e55c60c7059d407cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:26.431ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}I_{x}&={\frac {5{\sqrt {3}}}{16}}a^{4}\approx 0.54126a^{4}\\[3pt]I_{y}&={\frac {5{\sqrt {3}}}{16}}a^{4}\approx 0.54126a^{4}\end{aligned}}}"></span><sup id="cite_ref-roark_6-2" class="reference"><a href="#cite_note-roark-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></td> <td>The result is valid for both a horizontal and a vertical axis through the centroid, and therefore is also valid for an axis with arbitrary direction that passes through the origin. <p>This holds true for all <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygons</a>. </p> </td></tr> <tr> <td>A filled <a href="/wiki/Regular_octagon" class="mw-redirect" title="Regular octagon">regular octagon</a> with a side length of <i>a</i></td> <td><span typeof="mw:File"><a href="/wiki/File:Moment_of_area_of_a_regular_octagon.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Moment_of_area_of_a_regular_octagon.svg/230px-Moment_of_area_of_a_regular_octagon.svg.png" decoding="async" width="230" height="230" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Moment_of_area_of_a_regular_octagon.svg/345px-Moment_of_area_of_a_regular_octagon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/Moment_of_area_of_a_regular_octagon.svg/460px-Moment_of_area_of_a_regular_octagon.svg.png 2x" data-file-width="341" data-file-height="341" /></a></span> </td> <td><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 {\begin{aligned}I_{x}&={\frac {11+8{\sqrt {2}}}{12}}a^{4}\approx 1.85947a^{4}\\[3pt]I_{y}&={\frac {11+8{\sqrt {2}}}{12}}a^{4}\approx 1.85947a^{4}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.6em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>11</mn> <mo>+</mo> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mn>12</mn> </mfrac> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>≈</mo> <mn>1.85947</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>11</mn> <mo>+</mo> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mn>12</mn> </mfrac> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>≈</mo> <mn>1.85947</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I_{x}&={\frac {11+8{\sqrt {2}}}{12}}a^{4}\approx 1.85947a^{4}\\[3pt]I_{y}&={\frac {11+8{\sqrt {2}}}{12}}a^{4}\approx 1.85947a^{4}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1de48b68d2c6b13a938881757f6df6daa2dbbd3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:31.596ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}I_{x}&={\frac {11+8{\sqrt {2}}}{12}}a^{4}\approx 1.85947a^{4}\\[3pt]I_{y}&={\frac {11+8{\sqrt {2}}}{12}}a^{4}\approx 1.85947a^{4}\end{aligned}}}"></span><sup id="cite_ref-roark_6-3" class="reference"><a href="#cite_note-roark-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></td> <td>The result is valid for both a horizontal and a vertical axis through the centroid, and therefore is also valid for an axis with arbitrary direction that passes through the origin. <p>This holds true for all <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygons</a>. </p> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Parallel_axis_theorem">Parallel axis theorem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_second_moments_of_area&action=edit&section=2" title="Edit section: Parallel axis theorem"><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:Parallel_axis_theorem.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Parallel_axis_theorem.svg/220px-Parallel_axis_theorem.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Parallel_axis_theorem.svg/330px-Parallel_axis_theorem.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Parallel_axis_theorem.svg/440px-Parallel_axis_theorem.svg.png 2x" data-file-width="341" data-file-height="341" /></a><figcaption></figcaption></figure> <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">Main article: <a href="/wiki/Parallel_axis_theorem" title="Parallel axis theorem">Parallel axis theorem</a></div> <p>The parallel axis theorem can be used to determine the second moment of area of a rigid body about any axis, given the body's second moment of area about a parallel axis through the body's centroid, the area of the cross section, and the perpendicular distance (<i>d</i>) between the axes. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{x'}=I_{x}+Ad^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>x</mi> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mi>A</mi> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{x'}=I_{x}+Ad^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8087626f6ade20cc4fff661d5fe534c22d830db" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.877ex; height:3.009ex;" alt="{\displaystyle I_{x'}=I_{x}+Ad^{2}}"></span> </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=List_of_second_moments_of_area&action=edit&section=3" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/List_of_moments_of_inertia" title="List of moments of inertia">List of moments of inertia</a></li> <li><a href="/wiki/List_of_centroids" title="List of centroids">List of centroids</a></li> <li><a href="/wiki/Second_polar_moment_of_area" title="Second polar moment of area">Second polar moment of area</a></li></ul> <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=List_of_second_moments_of_area&action=edit&section=4" 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"> <div class="mw-references-wrap"><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 class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.efunda.com/math/areas/Circle.cfm">"Circle"</a>. eFunda<span class="reference-accessdate">. Retrieved <span class="nowrap">2006-12-30</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Circle&rft.pub=eFunda&rft_id=http%3A%2F%2Fwww.efunda.com%2Fmath%2Fareas%2FCircle.cfm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+second+moments+of+area" class="Z3988"></span></span> </li> <li id="cite_note-semicircle-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-semicircle_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-semicircle_2-1"><sup><i><b>b</b></i></sup></a></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.efunda.com/math/areas/CircleHalf.cfm">"Circular Half"</a>. eFunda<span class="reference-accessdate">. Retrieved <span class="nowrap">2006-12-30</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Circular+Half&rft.pub=eFunda&rft_id=http%3A%2F%2Fwww.efunda.com%2Fmath%2Fareas%2FCircleHalf.cfm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+second+moments+of+area" class="Z3988"></span></span> </li> <li id="cite_note-quartercircle-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-quartercircle_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-quartercircle_3-1"><sup><i><b>b</b></i></sup></a></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.efunda.com/math/areas/CircleQuarter.cfm">"Quarter Circle"</a>. eFunda<span class="reference-accessdate">. Retrieved <span class="nowrap">2006-12-30</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Quarter+Circle&rft.pub=eFunda&rft_id=http%3A%2F%2Fwww.efunda.com%2Fmath%2Fareas%2FCircleQuarter.cfm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+second+moments+of+area" class="Z3988"></span></span> </li> <li id="cite_note-rect-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-rect_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-rect_4-1"><sup><i><b>b</b></i></sup></a></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.efunda.com/math/areas/rectangle.cfm">"Rectangular area"</a>. eFunda<span class="reference-accessdate">. Retrieved <span class="nowrap">2006-12-30</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Rectangular+area&rft.pub=eFunda&rft_id=http%3A%2F%2Fwww.efunda.com%2Fmath%2Fareas%2Frectangle.cfm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+second+moments+of+area" class="Z3988"></span></span> </li> <li id="cite_note-tri-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-tri_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-tri_5-1"><sup><i><b>b</b></i></sup></a></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.efunda.com/math/areas/triangle.cfm">"Triangular area"</a>. eFunda<span class="reference-accessdate">. Retrieved <span class="nowrap">2006-12-30</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Triangular+area&rft.pub=eFunda&rft_id=http%3A%2F%2Fwww.efunda.com%2Fmath%2Fareas%2Ftriangle.cfm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+second+moments+of+area" class="Z3988"></span></span> </li> <li id="cite_note-roark-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-roark_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-roark_6-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-roark_6-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-roark_6-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYoung,_Warren_C;_Budynas,_Richard_G." class="citation book cs1">Young, Warren C; Budynas, Richard G. "Appendix A: Properties of a Plane Area". <a rel="nofollow" class="external text" href="http://nguyen.hong.hai.free.fr/EBOOKS/SCIENCE%20AND%20ENGINEERING/MECANIQUE/THEORIE%20DE%20BASE/Roark's%20Formulas%20For%20Stress%20And%20Strain.pdf"><i>Roark's Formulas for Stress and Strain. Seventh Edition</i></a> <span class="cs1-format">(PDF)</span>. pp. 802–812<span class="reference-accessdate">. Retrieved <span class="nowrap">23 December</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Appendix+A%3A+Properties+of+a+Plane+Area&rft.btitle=Roark%27s+Formulas+for+Stress+and+Strain.+Seventh+Edition.&rft.pages=802-812&rft.au=Young%2C+Warren+C%3B+Budynas%2C+Richard+G.&rft_id=http%3A%2F%2Fnguyen.hong.hai.free.fr%2FEBOOKS%2FSCIENCE%2520AND%2520ENGINEERING%2FMECANIQUE%2FTHEORIE%2520DE%2520BASE%2FRoark%27s%2520Formulas%2520For%2520Stress%2520And%2520Strain.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+second+moments+of+area" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=List_of_second_moments_of_area&oldid=1215767808">https://en.wikipedia.org/w/index.php?title=List_of_second_moments_of_area&oldid=1215767808</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Mechanics" title="Category:Mechanics">Mechanics</a></li><li><a href="/wiki/Category:Physics-related_lists" title="Category:Physics-related lists">Physics-related lists</a></li><li><a href="/wiki/Category:Moment_(physics)" title="Category:Moment (physics)">Moment (physics)</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">CS1 maint: multiple names: authors list</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 27 March 2024, at 00:20<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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List of second moments of area - Wikipedia