Perimeter | Brilliant Math & Science Wiki

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</div> <div class="alternative"> <p> Already have an account? <a href="https://brilliant.org/account/login/?next=/wiki/perimeter/" class="ax-click" data-ax-id="clicked_signup_modal_login" data-ax-type="link"> Log in here. </a> </p> </div> </div> </div> <div class="wiki-top-editors" id="cmp_wiki_top_editors_id"> <b>Mei Li</b>, <b>Ariel Gershon</b>, <b>Ashish Menon</b>, and <div class="dropdown tipsy"> <button class="btn-link dropdown-toggle" data-toggle="dropdown"> 10 others </button> <ul class="dropdown-menu"> <li> <b>Ram Mohith</b> </li> <li> <b>Prince Loomba</b> </li> <li> <b>Mira B</b> </li> <li> <b>Geoff Pilling</b> </li> <li> <b>Rex Holmes</b> </li> <li> <b>Worranat Pakornrat</b> </li> <li> <b>Jonathan Xian Kuan Chee</b> </li> <li> <b>A Former Brilliant Member</b> </li> <li> <b>Eli Ross</b> </li> <li> <b>Jimin Khim</b> </li> </ul> </div> contributed </div> <div id="wiki-main" data-controller="app/newsfeed:feed"> <div class="summary-container" id="cmp_wiki_canonical_page_id"> <div class="summary wiki-content" data-controller="app/wiki:summary,app/zoomable:images" data-cmp-url="/wiki/perimeter/" data-page-key="wiki_canonical_page" data-cmp-key="wiki_canonical_page"> <div class="section collapsed" id="section-pre-header-section"><div class="section-container"><p>The <strong>perimeter</strong> of a two-dimensional figure is the length of the boundary of the figure. If the figure is a <a href="/wiki/polygons/" class="wiki_link" title="polygon"target="_blank">polygon</a> such as a <a href="/wiki/triangles/" class="wiki_link" title="triangle"target="_blank">triangle</a>, <a href="/wiki/properties-of-squares/" class="wiki_link" title="square"target="_blank">square</a>, <a href="/wiki/properties-of-rectangles/" class="wiki_link" title="rectangle"target="_blank">rectangle</a>, <a href="/wiki/pentagons/" class="wiki_link" title="pentagon"target="_blank">pentagon</a>, etc., then the perimeter is the sum of the edge lengths of the polygon. For example, a square with side length 3 has perimeter $3+3+3+3 = 12.$ <span class="image-caption center"> <img src="http://g.recordit.co/gjewnkRtzp.gif" srcset="http://g.recordit.co/gjewnkRtzp.gif 1x" alt="" /> </span></p> <p><span class="image-caption right"> <img src="https://ds055uzetaobb.cloudfront.net/brioche/embedded_uploads/PVuiNp1KiU.png?width=1200" srcset="https://ds055uzetaobb.cloudfront.net/brioche/embedded_uploads/PVuiNp1KiU.png?width=1200 1x,https://ds055uzetaobb.cloudfront.net/brioche/embedded_uploads/PVuiNp1KiU.png?width=2400 2x,https://ds055uzetaobb.cloudfront.net/brioche/embedded_uploads/PVuiNp1KiU.png?width=3600 3x" alt="" /> </span> Any closed figure has a perimeter, such as the <a href="/wiki/irregular-polygons/" class="wiki_link" title="irregular polygon"target="_blank">irregular polygon</a> to the right with perimeter $1+1+2+ 3= 7.$ Some shapes do not have a finite number of sides, so calculating the perimeter can be less straighforward. For example, the perimeter (or <a href="/wiki/circles-circumference/" class="wiki_link" title="circumference"target="_blank">circumference</a>) of a <a href="/wiki/circles/" class="wiki_link" title="circle"target="_blank">circle</a> is $2\pi r,$ where $r$ is the <a href="/wiki/circles-radius-and-diameter/" class="wiki_link" title="radius"target="_blank">radius</a> of the circle. Fascinatingly, some shapes (such as certain <a href="/wiki/fractals/" class="wiki_link" title="fractals"target="_blank">fractals</a>) have <a href="/wiki/infinity/" class="wiki_link" title="infinite"target="_blank">infinite</a> perimeter, despite having a finite area. </p> </div> </div> <div class="toc wiki-toc"> <h4>Contents</h4> <ul class="unstyled"> <li> <a href="#perimeter-of-regular-polygons">Perimeter of Regular Polygons</a> </li> <li> <a href="#perimeter-of-irregular-polygons">Perimeter of Irregular Polygons</a> </li> <li> <a href="#perimeter-of-other-shapes-circles-fractals-etc">Perimeter of Other Shapes (Circles, Fractals, etc.)</a> </li> <li> <a href="#perimeter-using-polar-coordinates">Perimeter using Polar Coordinates</a> </li> <li> <a href="#problem-solving">Problem Solving</a> </li> <li> <a href="#real-life-applications">Real-life Applications</a> </li> </ul> </div> <div id="perimeter-of-regular-polygons" class="anchor skill-heading collapsed" data-controller="app/wiki:expandOrCollapse"> <header class="section-header"> <span class="css-sprite-chevrons chevron"></span> <h2>Perimeter of Regular Polygons</h2> </header> </div> <div class="section collapsed" id="section-perimeter-of-regular-polygons"><div class="section-container"> <blockquote class="example"><p> What is the perimeter of a square with side length 6?</p> <hr /> <p>Since a square has four sides of equal length and the perimeter of a square is the sum of its side lengths, the perimeter of the square is</p> <p>\[ 6 + 6 + 6 + 6 = 24. \ _\square \]</p> <p>You can also do $6 \times 4$ on the count that it is a square. </p></blockquote> <blockquote class="example"><p> What is the perimeter of a regular pentagon whose side lengths are all 6?</p> <hr /> <p>Since a regular pentagon has five sides and all its side lengths are $6$, the perimeter of the pentagon is</p> <p>\[ 6 + 6 + 6 + 6 + 6= 5 \cdot 6 = 30 . \ _\square \] </blockquote> </p> <p><div class="problem-modal-container anchor" id="problem-minimizing-perimeter-of-a-triangle"> <div class="wiki-problem" data-controller="app/wiki:wikiShowProblemAnswer"> <div class="problem-container" > <div class="answer-container solv-details"> <button class="btn reveal-solution-btn">Reveal the answer</button> </div> <div class="question-container"> <p>Find the minimum perimeter of a triangle whose area is $ \frac{\sqrt{3}}{4} $.</p> </div> <div class="solution-container hide"> <br><br> The correct answer is: <strong>3</strong> </div> </div> </div> </div> </p> </div> </div> <div id="perimeter-of-irregular-polygons" class="anchor skill-heading collapsed" data-controller="app/wiki:expandOrCollapse"> <header class="section-header"> <span class="css-sprite-chevrons chevron"></span> <h2>Perimeter of Irregular Polygons</h2> </header> </div> <div class="section collapsed" id="section-perimeter-of-irregular-polygons"><div class="section-container"> <blockquote class="example"><p> What is the perimeter of a rectangle with width $4$ and height $7?$</p> <hr /> <p>Since a rectangle has two sides with equal width and two sides with equal height, and the perimeter of a rectangle is the sum of its side lengths, the perimeter of the rectangle is</p> <p>\[ 4 + 4 + 7 + 7 = 2(4) + 2(7) = 8 + 14 = 22 . \ _\square \] </p></blockquote> <blockquote class="example"><p> Find the perimeter of a triangle $ABC$ with $AB= 3\text{ cm}, BC=4\text{ cm},$ and $\angle B= 90^{\circ}.$</p> <hr /> <p>First, use the Pythagorean theorem to find the length of $AC:$ </p> <p>\[AC^{2}=AB^{2}+BC^{2}=3^{2}+4^{2} \implies AC=5.\]</p> <p>Then add $AB,BC,CA$ to get the perimeter, which is</p> <p>\[3+4+5 =12\text{ (cm)}.\ _\square\] </p></blockquote> <blockquote class="example"><p> Suppose an ant walks along the boundary of a triangle with side lengths $4, 6,$ and $7$. If the ant starts at one vertex of the triangle and makes $3$ complete trips around the triangle boundary to return to its starting point, what is the total distance traveled by the ant?</p> <hr /> <p>Since the perimeter is the sum of edge lengths, the perimeter of the triangle is $4 + 6 + 7 = 17$. Since the ant makes $3$ complete trips around the triangle boundary, the total distance traveled by the ant is</p> <p>\[ 17 \times 3 = 51. \ _\square \] </blockquote> </p> <blockquote class="example"><p> <br /> <span class="image-caption center"> <img src="https://ds055uzetaobb.cloudfront.net/brioche/uploads/vaBeU3tnuQ-img_20160521_100512.jpg?width=1200" srcset="https://ds055uzetaobb.cloudfront.net/brioche/uploads/vaBeU3tnuQ-img_20160521_100512.jpg?width=1200 1x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/vaBeU3tnuQ-img_20160521_100512.jpg?width=2400 2x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/vaBeU3tnuQ-img_20160521_100512.jpg?width=3600 3x" alt="" /> </span> <br /> Find the perimeter of the given figure. </p> <hr /> <p>We have</p> <p>\[\begin{align} <br /> \text{Perimeter} & = \text{Sum of side lengths}\\ & = 9 + 8 + 7 + 3 + 4 + 5\\ & = 36.\ _\square \end{align}\] <br /> </blockquote> </p> <p><div class="problem-modal-container anchor" id="problem-a-plane-triangle"> <div class="wiki-problem" data-controller="app/wiki:wikiShowProblemAnswer"> <div class="problem-container" > <div class="answer-container solv-details"> <button class="btn reveal-solution-btn">Reveal the answer</button> </div> <div class="question-container"> <p><span class="image-caption center"> <img src="https://ds055uzetaobb.cloudfront.net/brioche/solvable/images/7516_diagramd-hzgX1UL7ol.png?width=1200" srcset="https://ds055uzetaobb.cloudfront.net/brioche/solvable/images/7516_diagramd-hzgX1UL7ol.png?width=1200 1x,https://ds055uzetaobb.cloudfront.net/brioche/solvable/images/7516_diagramd-hzgX1UL7ol.png?width=2400 2x,https://ds055uzetaobb.cloudfront.net/brioche/solvable/images/7516_diagramd-hzgX1UL7ol.png?width=3600 3x" alt="" /> </span></p> <p>Let there be a triangle drawn on a Cartesian plane with vertices at coordinates $(-3,0), (1,-3), (4,1).$ Find the perimeter of this triangle to the nearest integer.</p> </div> <div class="solution-container hide"> <br><br> The correct answer is: <strong>17</strong> </div> </div> </div> </div> </p> </div> </div> <div id="perimeter-of-other-shapes-circles-fractals-etc" class="anchor skill-heading collapsed" data-controller="app/wiki:expandOrCollapse"> <header class="section-header"> <span class="css-sprite-chevrons chevron"></span> <h2>Perimeter of Other Shapes (Circles, Fractals, etc.)</h2> </header> </div> <div class="section collapsed" id="section-perimeter-of-other-shapes-circles-fractals-etc"><div class="section-container"> <p>The perimeter of a circle with radius $r$ is $2\pi r$. The perimeter of an arc of a circle subtending angle $\theta$ (in degrees) at the center is $2\pi r \times \frac {\theta}{360}$.</p> <blockquote class="example"><p> What is the perimeter (circumference) of a circle of diameter $14\text{ cm}?$ </p> <hr /> <p>We have</p> <p>\[\begin{align} <br /> \text{Radius} & = \dfrac{\text{Diameter}}{2}\\ & = \dfrac{14}{2}\\ & = 7\text{ (cm)}\\ \\ \text{Circumference} & = 2脳 \pi 脳 r\\ & = 2 脳 \dfrac{22}{7} 脳 7\\ & = 44\text{ (cm)}.\ _\square \end{align}\] <br /> </blockquote> </p> <blockquote class="example"><p> Find the perimeter of a semicircle of radius $21$ cm. </p> <hr /> <p>We have</p> <p>\[\begin{align} <br /> \text{Perimeter of a semicircle} & = \left( \dfrac{1}{2} 脳 2 脳 \pi 脳 r \right) + 2r\\ & = \left( \dfrac{1}{2} 脳 2脳 \dfrac{22}{7} 脳 21\right) + 2脳21\\ & = 66 + 42\\ & = 106\text{ (cm)}.\ _\square \end{align}\] <br /> </blockquote> </p> <p><div class="problem-modal-container anchor" id="problem-approximating-pi-with-hexagons"> <div class="wiki-problem" data-controller="app/wiki:wikiShowProblemAnswer"> <div class="problem-container" > <div class="answer-container solv-details"> <button class="btn reveal-solution-btn">Reveal the answer</button> </div> <div class="question-container"> <p><span class="image-caption right"> <img src="https://ds055uzetaobb.cloudfront.net/brioche/uploads/UMXLWAaYUL-hexagon-pi-estimate-1.png?width=1200" srcset="https://ds055uzetaobb.cloudfront.net/brioche/uploads/UMXLWAaYUL-hexagon-pi-estimate-1.png?width=1200 1x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/UMXLWAaYUL-hexagon-pi-estimate-1.png?width=2400 2x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/UMXLWAaYUL-hexagon-pi-estimate-1.png?width=3600 3x" alt="" /> </span></p> <p>In the figure to the right, a circle is circumscribed around a regular hexagon, and the same circle is inscribed within another regular hexagon.</p> <p>Let $P_1$ be the perimeter of the larger hexagon, let $P_2$ be the perimeter of the smaller hexagon, and let $C$ be the circumference of the circle.</p> <p>The circumference of the circle can be approximated by finding the mean of the two perimeters:</p> <p>\[C\approx \frac{P_1+P_2}{2}.\]</p> <p>If $\pi$ is approximated using the approximation for circumference above, then $\pi\approx\frac{a}{b}+\sqrt{c}$, where $a, b, c$ are positive integers, $a$ and $b$ are coprime, and $c$ is square-free.</p> <p>Find $a+b+c$.</p> </div> <div class="solution-container hide"> <br><br> The correct answer is: <strong>8</strong> </div> </div> </div> </div> </p> </div> </div> <div id="perimeter-using-polar-coordinates" class="anchor skill-heading collapsed" data-controller="app/wiki:expandOrCollapse"> <header class="section-header"> <span class="css-sprite-chevrons chevron"></span> <h2>Perimeter using Polar Coordinates</h2> </header> </div> <div class="section collapsed" id="section-perimeter-using-polar-coordinates"><div class="section-container"> <p>We can calculate the perimeter of an enclosed figure if its graph on a Cartesian plane is a function in polar coordinates. Specifically, suppose the figure has the equation $r = f(\theta)$ such that $f$ is a continuous, nonnegative function for $\theta \in [0, 2\pi]$ and $f(0) = f(2\pi)$. Then its perimeter is given by the following formula:</p> <p>\[P = \int_{0}^{2\pi} \sqrt{r^2 + \left(\dfrac{dr}{d\theta}\right)^2} d\theta.\]</p> <blockquote class="example"><p> Suppose we want to calculate the circumference of a circle with radius $r$. Then $r$ is a constant, and its derivative is $0$. So the formula gives us</p> <p>\[P = \int_{0}^{2\pi} \sqrt{r^2 + 0^2} d\theta = \int_{0}^{2\pi} r d\theta = r(2\pi-0) = 2\pi r.\]</p> <p>Therefore the circumference is $2\pi r.$ </blockquote> </p> <p>Unfortunately, for other shapes, this formula usually leads to complicated integrals which often do not have a simple closed form.</p> </div> </div> <div id="problem-solving" class="anchor skill-heading collapsed" data-controller="app/wiki:expandOrCollapse"> <header class="section-header"> <span class="css-sprite-chevrons chevron"></span> <h2>Problem Solving</h2> </header> </div> <div class="section collapsed" id="section-problem-solving"><div class="section-container"> <p><div class="problem-modal-container anchor" id="problem-prime-square"> <div class="wiki-problem" data-controller="app/wiki:wikiShowProblemAnswer"> <div class="problem-container" > <div class="answer-container solv-details"> <button class="btn reveal-solution-btn">Reveal the answer</button> </div> <div class="question-container"> <p>A red rectangle is reshaped into a blue square, as shown below, such that both quadrilaterals have the same <a href="/wiki/perimeter/" class="wiki_link" title="perimeter"target="_blank">perimeter</a>. The area of the square is $4\text{ cm}^2$ larger than that of the rectangle.</p> <p>If all the sides of both quadrilaterals have <a href="/wiki/prime-numbers/" class="wiki_link" title="prime number"target="_blank">prime number</a> lengths $($in $\text{cm}),$ then what is the perimeter of the square? <span class="image-caption center"> <img src="https://ds055uzetaobb.cloudfront.net/brioche/uploads/dGiS97LVzN-prime-square.jpg?width=1200" srcset="https://ds055uzetaobb.cloudfront.net/brioche/uploads/dGiS97LVzN-prime-square.jpg?width=1200 1x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/dGiS97LVzN-prime-square.jpg?width=2400 2x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/dGiS97LVzN-prime-square.jpg?width=3600 3x" alt="" /> </span></p> </div> <div class="solution-container hide"> <br><br> The correct answer is: <strong>20</strong> </div> </div> </div> </div> <div class="problem-modal-container anchor" id="problem-perimeter-of-trapezium"> <div class="wiki-problem" data-controller="app/wiki:wikiShowProblemAnswer"> <div class="problem-container" > <div class="answer-container solv-details"> <button class="btn reveal-solution-btn">Reveal the answer</button> </div> <div class="question-container"> <p><span class="image-caption right"> <img src="https://ds055uzetaobb.cloudfront.net/brioche/uploads/yU5lToR6VD-images.png?width=1200" srcset="https://ds055uzetaobb.cloudfront.net/brioche/uploads/yU5lToR6VD-images.png?width=1200 1x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/yU5lToR6VD-images.png?width=2400 2x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/yU5lToR6VD-images.png?width=3600 3x" alt="" /> </span> If the altitude and perimeter of the trapezium are $3$ and $A+B \sqrt{2}+C \sqrt{3},$ respectively, find $A+B+C$.</p> </div> <div class="solution-container hide"> <br><br> The correct answer is: <strong>35</strong> </div> </div> </div> </div> <div class="problem-modal-container anchor" id="problem-perimeter-of-trapezium-again"> <div class="wiki-problem" data-controller="app/wiki:wikiShowProblemAnswer"> <div class="problem-container" > <div class="answer-container solv-details"> <button class="btn reveal-solution-btn">Reveal the answer</button> </div> <div class="question-container"> <p><span class="image-caption left"> <img src="https://ds055uzetaobb.cloudfront.net/brioche/uploads/zsiEwSRMpT-s1q8a.png?width=1200" srcset="https://ds055uzetaobb.cloudfront.net/brioche/uploads/zsiEwSRMpT-s1q8a.png?width=1200 1x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/zsiEwSRMpT-s1q8a.png?width=2400 2x,https://ds055uzetaobb.cloudfront.net/brioche/uploads/zsiEwSRMpT-s1q8a.png?width=3600 3x" alt="" /> </span> Find the perimeter of this figure in $\si{\cm}.$</p> </div> <div class="solution-container hide"> <br><br> The correct answer is: <strong>44</strong> </div> </div> </div> </div> </p> </div> </div> <div id="real-life-applications" class="anchor skill-heading collapsed" data-controller="app/wiki:expandOrCollapse"> <header class="section-header"> <span class="css-sprite-chevrons chevron"></span> <h2>Real-life Applications</h2> </header> </div> <div class="section collapsed" id="section-real-life-applications"><div class="section-container"> <p>In this section, we will learn about applications of the concept of perimeter in real life. They are mostly of fencing-type problems. Here is an example:</p> <blockquote class="example"><p> Mr. Antony has a rectangular garden of area $630\text{ m}^2$ and length $42\text{ m}.$ To save his garden from domestic animals grazing them, he wants to fence his whole garden. Find much Mr. Antony has to spend for fencing his field at the rate of $3 per meter?</p> <hr /> <p>Given, the area of the rectangle $A = 630\text{ m}^2$ and its length $l = 42\text{ m},$ we know that </p> <p>\[A = l \times b \implies b = \dfrac{A}{l} = \dfrac{630}{42} = 15\ (\text{m}).\]</p> <p>Now, let's find perimeter: $P = 2(l + b) = 2(42 + 15) = 114\ (\text{m}).$</p> <p>Since the cost of fencing per meter is $3, the total cost of fencing the whole feild is $ 3 \times P = 3 \times 114 = 342$ dollars. $_\square$ </p></blockquote> </div> </div> </div> </div> </div> <div class="wiki-self-citation" data-controller="app/wiki:getCitationTime"> <strong>Cite as:</strong> Perimeter. <em>Brilliant.org</em>. 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Perimeter | Brilliant Math & Science Wiki