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122 lines
6.2 KiB
HTML
122 lines
6.2 KiB
HTML
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<!-- Mirrored from www.mathsisfun.com/geometry/circle-area-by-sectors.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:37:25 GMT -->
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<title>Area of a Circle by Cutting into Sectors</title>
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<h1 align="center">Area of a Circle by Cutting into Sectors</h1>
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<p> </p>
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<p>Here is a way to find the formula for the area of a circle:</p>
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<p style="float:left; margin: 0 20px 0 0;"><img src="../activity/images/pi-1.gif" width="178" height="183" alt="circle 12 sectors" /></p>
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<p> </p>
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<p>Cut a circle into equal sectors (12 in this example)</p>
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<div class="so"></div>
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<p>Divide just one of the sectors into two equal parts. We now have thirteen sectors – number them 1 to 13:</p>
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<p align="center"><img src="../activity/images/pi-approx-13.svg" alt="circle 13 including 2 half slices" /></p>
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<p>Rearrange the 13 sectors like this:</p>
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<p align="center"><img src="../activity/images/pi-4.gif" width="282" height="105" alt="sectors laid out like rectangle" /></p>
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<p>Which resembles a rectangle:</p>
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<p align="center"><img src="../activity/images/pi-4a.gif" width="278" height="100" alt="sectors with rectangle on top" /></p>
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<p>What are the (approximate) height and width of the rectangle? </p>
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<div class="dotpoint">
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<p>The <b>height</b> is the circle's <b>radius</b>: just look at sectors 1 and 13 above. When they were in the circle they were "radius" high.</p>
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</div>
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<div class="dotpoint">
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<p>The <b>width</b> (actually one "bumpy" edge) is half of the curved parts around the circle ... in other words it is about <b>half the circumference</b> of the circle. </p>
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<p>We know that:</p>
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<p align="center" class="larger">Circumference = 2 × <span class="times">π</span> × radius</p>
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<p>And so the width is about:</p>
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<p align="center"><span class="larger">Half the Circumference = <span class="times">π</span> × radius </span></p>
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</div>
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<p>And so we have (approximately):</p>
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<table border="0" align="center">
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<tbody><tr>
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<td><img src="../activity/images/pi-5.gif" alt="rectangle is (pi x radius) by radius" width="285" height="104"></td>
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<td><span class="larger">radius</span></td>
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</tr>
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<tr align="center">
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<td><span class="larger"><span class="times">π</span> ×
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radius</span></td>
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<td> </td>
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</tr>
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</tbody></table>
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<p>Now we just multply the width by the height to find the area of the rectangle:</p>
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<p align="center"><span class="larger">Area = (<span class="times">π</span> × radius) × (radius) </span></p>
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<p align="center"><span class="larger">= <span class="times">π</span> × radius<sup>2</sup></span></p>
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<div class="center80">
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<p>Note: The rectangle and the "bumpy edged shape" made by the sectors are not an exact match. </p>
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<p>But we could get a better result if we divided the circle into 25 sectors (23 with an angle of 15° and 2 with an angle of 7.5°).</p>
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<p>
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And the more we divided the circle up, the closer we get to being exactly right.</p>
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</div>
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<h2>Conclusion</h2>
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<p align="center"><span class="larger">Area of Circle = <span class="times">π </span>r<sup>2</sup></span></p>
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<p> </p>
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<div class="related">
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<a href="circle.html">Circle</a><a href="circle-area-lines.html">Circle Area by Lines</a>
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<a href="../activity/pi-approximation.html">Pi Approximation Activity</a><a href="index.html">Geometry Index</a>
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Copyright © 2017 MathsIsFun.com
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