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<h1 class="center">Perimeter of an Ellipse</h1>
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<p>On the <a href="ellipse.html">Ellipse</a> page we looked at the definition and some of the simple properties of the ellipse, but here we look at how to more accurately calculate its perimeter.</p>
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<h2>Perimeter</h2>
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<p>Rather strangely, the perimeter of an ellipse is <b>very difficult to calculate</b>!</p>
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<p>There are many formulas, here are some interesting ones. (Also see <a href="#tool">Calculation Tool</a> below.)</p>
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<h2>First Measure Your Ellipse!</h2>
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<p class="center"><img src="images/ellipse-axes.svg" alt="ellipse major and minor axes" height="143" width="287"></p>
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<p class="center"><b>a</b> and <b>b</b> are measured <b>from the center</b>, so they are like "radius" measures.</p>
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<p> </p>
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<h3>Approximation 1</h3>
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<p>This approximation is within about 5% of the true value, so long as <b>a</b> is not more than 3 times longer than <b>b</b> (in other words, the ellipse is not too "squashed"):</p>
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<div class="center large">p ≈ 2<span class="times">π</span> <span class="tall">√</span><span class="intbl"><span class="overline"><em>a<sup>2</sup>+b<sup>2</sup></em><strong>2</strong></span></span></div>
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<!-- p APR 2 PI SQR(a^2+b^2/2) -->
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<h3>Approximation 2</h3>
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<p>The famous Indian mathematician <b>Ramanujan</b> came up with this better approximation:</p>
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<p class="center "><img src="images/ellipse-perim-2.gif" class="bgwhite" alt="ellipse perimeter approx pi [ 3(a+b) - sqrt((3a+b)(a+3b))]" height="41" width="310"></p>
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<h3>Approximation 3</h3>
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<p><b>Ramanujan</b> also came up with this one. First we calculate "h":</p>
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<div class="center large">h = <span class="intbl"><em>(a − b)<sup>2</sup></em><strong>(a + b)<sup>2</sup></strong></span></div>
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<!-- h = (a-b)^2/(a+b)^2 -->
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<p>Then use it here:</p>
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<div class="center large">p ≈ <span class="times">π</span>(a+b) <span class="tall">(</span> 1 + <span class="intbl"><em>3h</em><strong>10 + <span style="font-size:110%;">√</span>(4−3h)</strong></span> <span class="tall">)</span></div>
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<!-- p APR PI(a+b) 1+3h/10+SQR(4-3h) -->
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<h3>Infinite Series 1</h3>
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<p>This is an <b>exact formula</b>, but it needs an "infinite series" of calculations to be exact, so in practice we still only get an approximation.</p>
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<p>First we calculate <span class="large">e</span> (the "<a href="eccentricity.html">eccentricity</a>", <b>not</b> <a href="../numbers/e-eulers-number.html">Euler's number "e"</a>):</p>
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<div class="center large">e = <span class="intbl"><em><span style="font-size:120%;">√</span><span class="overline">a<sup>2</sup> − b<sup>2</sup></span></em><strong>a</strong></span></div>
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<!-- e = SQR(a^2-b^2)/a -->
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<p>Then use this "infinite sum" formula:</p>
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<p class="center"><img src="images/ellipse-perim-4.gif" alt="ellipse perimeter approx 2a pi [ 1 - sigma i=1 to infinity of ( (2i)!^2/(i!2^i)^4 times e^21/(2i-1))]" class="bgwhite" height="51" width="295"></p>
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<p>Which may look complicated, but expands like this:</p>
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<p class="center"><img src="images/ellipse-perim-5.gif" alt="ellipse perimeter approx 2a pi [ 1 - (1/2)^2 e^2 - (1x3/2x4)^2 e^4 /3 - (1x3x5/2x4x6)^2 e^6 /5 - ... ]" class="bgwhite" height="51" width="497"></p>
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<p>The terms continue on infinitely, and unfortunately we must calculate a lot of terms to get a reasonably close answer.</p>
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<h3>Infinite Series 2</h3>
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<p>But my favorite <b>exact formula</b> (because it gives a very close answer after only a few terms) is as follows:</p>
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<p>First we calculate "h":</p>
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<div class="center large">h = <span class="intbl"><em>(a − b)<sup>2</sup></em><strong>(a + b)<sup>2</sup></strong></span></div>
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<p></p>
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<p>Then use this "infinite sum" formula:</p>
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<p class="center"><img src="images/ellipse-perim-7.gif" alt="ellipse perimeter approx pi(a+b) sigma n=0 to infinity of (0.5 choose n)^2 h^n " class="bgwhite" height="54" width="219"></p>
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<p class="center">(Note: the <img src="images/combinations-half-n.gif" style="vertical-align:middle;" alt="combinations-half-n" class="bgwhite" height="30" width="25"> is the <a href="../combinatorics/combinations-permutations.html">Binomial Coefficient</a>
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with half-integer <a href="../numbers/factorial.html">factorials</a> ... wow!)</p>
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<p>It may look a bit scary, but it expands to this series of calculations:</p>
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<p class="center"><img src="images/ellipse-perim-8.gif" alt="ellipse perimeter approx pi(a+b) (1 + (1/4)h + (1/64)h^2 + (1/256)h^3 + ...)" class="bgwhite" height="43" width="372"></p>
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<p>The more terms we calculate, the more accurate it becomes (the next term is 25<b>h</b><sup>4</sup>/16384, which is getting quite small, and the next is 49<b>h</b><sup>5</sup>/65536, then 441<b>h</b><sup>6</sup>/1048576, then 1089<b>h</b><sup>7</sup>/4194304)</p>
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<h2>The Perfect Formula</h2>
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<p>There is a perfect formula using an <a href="../calculus/integration-introduction.html">integral</a>:</p>
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<div class="center larger">p = 4a
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<div class="intgl">
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<div class="to"><span class="intbl"><span class="times">π</span>/2</span></div>
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<div class="symb"></div>
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<div class="from">0</div>
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</div> √(1 − e<sup>2</sup> sin<sup>2</sup> θ) dθ</div>
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<p class="center"><i>(Note: e is the eccentricity from above)</i></p>
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<p>But calculating it needs an infinite amount of terms ("Infinite Series 1" above).</p>
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<h2>Comparing</h2>
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<p>Just for fun, I calculate the perimeter using the three approximation formulas, and the two exact formulas (but only the <b>first four terms, including the "1"</b>, so it is still just an approximation) for selected values of <b>a</b> and <b>b</b>:</p>
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<table width="100%">
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<tbody>
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<tr>
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<td style="text-align:right;"><br>
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</td>
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<td> </td>
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<td><b>Circle</b></td>
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<td><br>
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</td>
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<td><br>
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</td>
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<td><br>
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</td>
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<td><b>Lines</b></td></tr>
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<tr valign="middle">
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<td style="text-align:right;"> </td>
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<td> </td>
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<td><img src="images/ellipse-10-10.gif" alt="ellipse 10 10" height="56" width="58"></td>
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<td><img src="images/ellipse-10-5.gif" alt="ellipse 10 5" height="30" width="58"></td>
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<td><img src="images/ellipse-10-3.gif" alt="ellipse 10 3" height="20" width="58"></td>
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<td><img src="images/ellipse-10-1.gif" alt="ellipse 10 1" height="8" width="58"></td>
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<td><img src="images/ellipse-10-0.gif" alt="ellipse 10 0" height="5" width="57"></td>
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</tr>
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<tr>
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<td style="text-align:right;"><b>a:</b></td>
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<td> </td>
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<td><b>10</b></td>
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<td><b>10</b></td>
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<td><b>10</b></td>
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<td><b>10</b></td>
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<td><b>10</b></td></tr>
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<tr>
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<td style="text-align:right;"><b>b:</b></td>
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<td> </td>
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<td><b>10</b></td>
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<td><b>5</b></td>
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<td><b>3</b></td>
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<td><b>1</b></td>
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<td><b>0</b></td></tr>
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<tr>
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<td style="text-align:right;"><b>Approx 1:</b></td>
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<td> </td>
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<td>62.832</td>
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<td>49.673</td>
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<td>46.385</td>
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<td>44.65</td>
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<td>44.429</td></tr>
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<tr>
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<td style="text-align:right;"><b>Approx 2:</b></td>
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<td> </td>
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<td>62.832</td>
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<td>48.442</td>
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<td>43.857</td>
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<td>40.606</td>
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<td>39.834</td></tr>
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<tr>
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<td style="text-align:right;"><b>Approx 3:</b></td>
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<td> </td>
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<td>62.832</td>
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<td>48.442</td>
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<td>43.859</td>
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<td>40.639</td>
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<td>39.984</td></tr>
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<tr>
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<td style="text-align:right;"><b>Series 1:</b></td>
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<td> </td>
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<td>62.832</td>
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<td>48.876</td>
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<td>45.174</td>
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<td>43.204</td>
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<td>42.951</td></tr>
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<tr>
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<td style="text-align:right;"><b>Series 2:</b></td>
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<td> </td>
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<td>62.832</td>
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<td>48.442</td>
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<td>43.859</td>
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<td>40.623</td>
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<td>39.884</td></tr>
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<tr>
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<td style="text-align:right;"><b>Exact*:</b></td>
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<td> </td>
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<td><b>20<span class="times">π</span></b></td>
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<td><br>
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</td>
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<td><br>
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</td>
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<td><br>
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</td>
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<td><b>40</b></td></tr>
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</tbody></table>
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<p><br>
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<b>* Exact:</b></p>
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<ul>
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<li>When <b>a=b</b>, the ellipse is a circle, and the perimeter is <b>2<span class="times">π</span>a</b> (62.832... in our example).</li>
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<li>When <b>b=0</b> (the shape is really two lines back and forth) the perimeter is <b>4a</b> (40 in our example).</li>
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</ul>
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<p>They all get the perimeter of the circle correct, but only <b>Approx 2 and 3</b> and <b>Series 2</b> get close to the value of 40 for the extreme case of b=0.</p>
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<h2><a id="tool"></a>Ellipse Perimeter Calculations Tool</h2>
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<p>This tool does the calculations from above, but with more terms for the Series.</p>
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<div class="script" style="height: 360px;">
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images/ellipse-perim.js
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</div>
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<p> </p>
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<div class="related">
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<a href="ellipse.html">Ellipse</a>
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<a href="index.html">Geometry Index</a>
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<footer id="footer" class="centerfull noprint"></footer>
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<div id="copyrt">Copyright © 2022 Rod Pierce</div>
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