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149 lines
8.1 KiB
HTML
149 lines
8.1 KiB
HTML
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<title>Circle Sector and Segment</title>
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<h1 align="center">Circle Sector and Segment</h1>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/circle-slices.svg" alt="circle sector segment" /></p>
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<h2>Slices</h2>
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<p>There are two main "slices" of a <a href="circle.html">circle</a>:</p>
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<ul>
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<div class="bigul">
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<li>The "pizza" slice is called a <b>Sector</b>.</li>
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<li>And the <b>Segment</b>, which is cut from the circle by a "chord" (a line between two points on the circle).</li>
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</div>
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</ul>
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<h2>Try Them!</h2>
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<div class="simple">
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<table border="0" align="center">
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<tr align="center">
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<th>Sector</th>
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<th>Segment</th>
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</tr>
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<tr bgcolor="#FFFFFF">
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<td><iframe src="icircle-sector.html" scrolling="no" style="width:264px; height:264px; overflow:hidden; margin:auto; display:block; border: 1px solid blue; border-radius: 5px;"></iframe></td>
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<td><iframe src="icircle-segment.html" scrolling="no" style="width:264px; height:264px; overflow:hidden; margin:auto; display:block; border: 1px solid blue; border-radius: 5px;"></iframe></td>
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</tr>
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</table>
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</div>
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<h2> Common Sectors </h2>
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<p>The Quadrant and Semicircle are two special types of Sector:</p>
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<div style="float:left; margin: 0 10px 5px 0;">
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<p><img src="images/circle-semicircle.svg" alt="semicircle" /><br>
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Half a circle is <br>
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a <b>Semicircle.</b></p>
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</div>
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<div style="float:right; margin: 0 0 5px 10px;">
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<p class="right"><img src="images/circle-quadrant.svg" alt="quadrant" /><br>
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Quarter of a circle is<br>
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a <b>Quadrant</b>.</p>
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</div>
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<div style="clear:both"></div>
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<h2>Area of a Sector</h2>
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<p>You can work out the Area of a Sector by comparing its angle to the angle of a full circle.</p>
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<p class="center"><i>Note: we are using <a href="radians.html">radians</a> for the angles.</i></p>
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<p class="center"><img src="images/circle-sector-area.svg" alt="circular sector area" /></p>
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<p>This is the reasoning:</p><div class="tbl">
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<div class="row"><span class="left">A circle has an angle of 2<span class="times">π</span> and an Area of:</span><span class="right"><span class="times">π</span>r<sup>2</sup></span></div>
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<div class="row"><span class="left">A Sector has an angle of <span class="times">θ</span> <b>instead of</b> 2<span class="times">π</span> so its Area is : </span><span class="right"><span class="intbl"><em><span class="times">θ</span></em><strong>2<span class="times">π</span></strong></span> ×<span class="times"> π</span>r<sup>2</sup></span></div>
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<div class="row"><span class="left">Which can be simplified to:</span><span class="right"><span class="intbl"><em><span class="times">θ</span></em><strong>2</strong></span> × r<sup>2</sup></span></div>
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</div>
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<p> </p>
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<p align="center" class="large">Area of Sector = <span class="intbl">
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<em>θ</em>
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<strong>2</strong>
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</span> × r<sup>2</sup> <i>(when θ is in radians)</i></p>
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<p class="center large">Area of Sector = <span class="intbl">
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<em>θ × <span class="times">π</span></em>
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<strong>360</strong>
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</span> × r<sup>2</sup> <i>(when θ is in degrees) </i></p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/circle-segment-area.svg" alt="circular segment area" /></p>
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<h2>Area of Segment</h2>
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<p>The Area of a Segment is the area of a sector minus the triangular piece (shown in light blue here). </p>
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<p>There is a lengthy reason, but the result is a slight modification of the Sector formula:</p>
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<div style="clear:both"></div><p class="center large">Area of Segment = <span class="intbl">
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<em>θ − sin(θ)</em>
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<strong>2</strong>
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</span> × r<sup>2</sup> <i>(when θ is in radians)</i></p>
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<p class="center large">Area of Segment = ( <span class="intbl">
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<em>θ × <span class="times">π</span></em>
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<strong>360</strong>
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</span>
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− <span class="intbl"><em>sin(θ)</em><strong>2
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</strong></span>) × r<sup>2</sup> <i>(when θ is in degrees)</i></p>
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<p> </p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/circle-arc-length.svg" alt="circular sector arc length" />
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</p>
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<h2>Arc Length</h2>
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<p>The arc length (of a <b>Sector or Segment</b>) is:</p>
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<p class="center large">L = θ × r <i>(when θ is in radians)</i></p>
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<p class="center large">L = θ × <span class="intbl"><em><span class="times">π</span></em><strong>180</strong></span> × r <i>(when θ is in degrees)</i></p>
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<p> </p>
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<iframe src="icircle-arclen.html" scrolling="no" style="width:264px; height:264px; overflow:hidden; margin:auto; display:block; border: 1px solid blue; border-radius: 5px;"></iframe>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(770, 771, 772, 773, 1767, 1768, 1765, 1766, 3238, 3239);</script> </div>
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<div class="related"> <a href="../numbers/pi.html">Pi</a> <a href="plane.html">Plane</a> <a href="unit-circle.html">Unit Circle</a> <a href="plane-geometry.html">Plane Geometry</a> <a href="index.html">Geometry Index</a></div>
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