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400 lines
18 KiB
HTML
400 lines
18 KiB
HTML
<!doctype html>
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<html lang="en"><!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
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<!-- Mirrored from www.mathsisfun.com/geometry/regular-polygons.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:42:52 GMT -->
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<head>
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<title>Regular Polygons - Properties</title>
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<div id="content" role="main"><!-- #BeginEditable "Body" -->
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<h1 align="center">Properties of Regular Polygons</h1>
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<h2>Polygon</h2>
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<p>A <a href="polygons.html">polygon</a> is a <a href="plane.html">plane</a> shape (two-dimensional) with straight sides. Examples include triangles, quadrilaterals, pentagons, hexagons and so on.</p>
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<h2>Regular</h2>
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<table width="100%" border="0">
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<tr>
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<td>
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<p>A "Regular Polygon" has:</p>
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<ul>
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<div class="bigul">
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<li>all <b>sides</b> equal and </li>
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<li>all <b>angles</b> equal. </li>
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</div>
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</ul>
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<p>Otherwise it is<b> irregular</b>.</p>
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</td>
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<td>
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<table border="0" align="center">
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<tr align="center">
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<td><b><img src="images/pentagon-regular.svg" alt="pentagon regular" /></b></td>
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<td width="40"> </td>
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<td><b><img src="images/pentagon-irregular.svg" alt="irregular pentagon" /></b></td>
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</tr>
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<tr align="center">
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<td>Regular Pentagon</td>
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<td> </td>
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<td>Irregular Pentagon</td>
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</tr>
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</table>
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</td>
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</tr>
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</table>
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<p>Here we look at <b>Regular Polygons</b> only.</p>
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<h2>Properties</h2>
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<p>So what can we know about regular polygons? First of all, we can work out angles.</p>
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<table width="100%" border="0">
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<tr>
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<td><img src="images/exterior-angle.svg" alt="exterior angle" /></td>
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<td>
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<h2>Exterior Angle</h2>
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<p>The <a href="exterior-angles-polygons.html">Exterior Angle</a> is the angle between any side of a shape,
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<br /> and a line extended from the next side.</p>
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</td>
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</tr>
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</table>
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<div style="float:left; margin: 0 10px 5px 0;">
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<script type="text/javascript">
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exterioranglesMain();
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</script>
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</div>
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<p>All the Exterior Angles of a polygon add up to 360°, so:</p>
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<p class="center larger">Each exterior angle must be <b>360°/n </b></p>
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<p>(where <b>n</b> is the number of sides)</p>
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<p> </p>
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<p>Press play button to see.</p>
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<div style="clear:both"></div>
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<div class="example">
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<p style="float:right; margin: 10px 0 5px 10px;" class="center"><img src="images/external-angle.svg" alt="external angle of regular octagon" />
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<br> <i>Exterior Angle<br>(of a regular octagon)</i></p>
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<h3>Example: What is the exterior angle of a regular octagon?</h3>
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<p> </p>
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<p>An octagon has 8 sides, so:</p>
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<div class="tbl">
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<div class="row"><span class="left">Exterior angle =</span><span class="right">360° / n</span></div>
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<div class="row"><span class="left"> =</span><span class="right">360° / 8</span></div>
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<div class="row"><span class="left">=</span><span class="right">45°</span></div>
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</div>
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</div>
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<div style="clear:both"></div>
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<table width="100%" border="0">
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<tr>
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<td><img src="images/exterior-interior-angle.gif" width="200" height="103" alt="exterior interior angle" /></td>
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<td>
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<h2>Interior Angles </h2>
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<p>The <a href="interior-angles-polygons.html">Interior Angle</a> and Exterior Angle are measured from the same line, so they <b>add up to 180°</b>.</p>
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</td>
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</tr>
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</table>
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<p class="center larger">Interior Angle = 180° − Exterior Angle</p>
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<p>We know the<b> Exterior angle = 360°/n</b>, so:</p>
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<p class="center larger">Interior Angle = 180° − 360°/n </p>
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<div class="center80">
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<p>Which can be rearranged like this:</p><div class="tbl">
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<div class="row"><span class="left"> Interior Angle</span><span class="right">= 180° − 360°/n</span></div>
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<div class="row"><span class="left"> </span><span class="right">= (n × 180° / n) − (2 × 180° / n)</span></div>
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<div class="row"><span class="left"> </span><span class="right"> = (n−2) × 180°/n</span></div>
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</div>
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<p>So we also have this:</p>
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<p class="center larger">Interior Angle = (n−2) × 180° / n</p>
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</div>
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<p> </p>
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<div class="example">
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<h3>Example: What is the interior angle of a regular octagon?</h3>
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<p>A regular octagon has 8 sides, so:</p>
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<p class="center">Exterior Angle = 360<b>° </b>/ 8 = 45°</p>
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<p class="center"> Interior Angle = 180° − 45° = <b>135°</b> </p>
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<p class="center"><img src="images/internal-angle.svg" alt="internal angle of regular octagon" /><i><br>
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Interior Angle<br>
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(of a regular octagon)</i></p>
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<p>Or we could use: </p>
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<div class="tbl">
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<div class="row"><span class="left"> Interior Angle</span><span class="right">= (n−2) × 180° / n</span></div>
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<div class="row"><span class="left"> </span><span class="right"> = (8−2) × 180° / 8</span></div>
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<div class="row"><span class="left"> </span><span class="right"> = 6 × 180° / 8</span></div>
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<div class="row"><span class="left"> </span><span class="right"> = 135°</span></div>
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</div>
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</div>
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<div class="example">
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<h3>Example: What are the interior and exterior angles of a regular hexagon? </h3>
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<p style="float:left; margin: 10px;"><img src="images/regular-hexagon.svg" alt="regular hexagon" /></p>
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<p>A regular hexagon has 6 sides, so:</p>
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<p class="center">Exterior Angle = 360<b>° </b>/ 6 = 60°</p>
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<p class="center"> Interior Angle = 180<b>° − </b> 60° = <b>120°</b></p>
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</div>
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<br />
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<p>And now for some names:</p>
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<h2>"Circumcircle, Incircle, Radius and Apothem ..." </h2>
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<p>Sounds quite musical if you repeat it a few times, but they are just the names of the "outer" and "inner" circles (and each radius) that can be drawn on a polygon like this:</p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/apothem.svg" alt="apothem incircle, radius, circumcircle" /></p>
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<p> </p>
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<p>The "outside" circle is called a <b>circumcircle</b>, and it connects all vertices (corner points) of the polygon.</p>
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<p>The radius of the circumcircle is also the <b>radius</b> of the polygon.</p>
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<p> </p>
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<p>The "inside" circle is called an <b>incircle</b> and it just touches each side of the polygon at its midpoint.</p>
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<p>The radius of the incircle is the <b>apothem</b> of the polygon.</p>
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<p> </p>
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<p>(Not all polygons have those properties, but triangles and regular polygons do).</p>
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<h2>Breaking into Triangles</h2>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/hexagon-triangles.svg" alt="hexagon triangles side and apothem" /></p>
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<p>We can learn a lot about regular polygons by breaking them into triangles like this:</p>
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<p>Notice that:</p>
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<ul>
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<li>the "base" of the triangle is one side of the polygon.</li>
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<li>the "height" of the triangle is the "Apothem" of the polygon</li>
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</ul>
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<p>Now, the <a href="../area.html">area of a triangle</a> is half of the base times height, so:</p>
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<p class="center">Area of one triangle = base × height / 2 = side × apothem / 2</p>
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<p>To get the area of the whole polygon, just add up the areas of all the little triangles ("n" of them): </p>
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<p class="center">Area of Polygon = <b>n</b> × side × apothem / 2</p>
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<p>And since the perimeter is all the sides = n × side, we get: </p>
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<p class="center larger">Area of Polygon = perimeter × apothem / 2</p>
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<h2>A Smaller Triangle</h2>
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<p>By cutting the triangle in half we get this:</p>
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<p class="center"><img src="images/regular-polygon-sector.svg" alt="regular polygon sector" /><i><br>
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<br>
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(Note: The angles are in <a href="radians.html">radians</a>, not <a href="degrees.html">degrees</a>)</i></p>
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<p> </p>
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<p>The small triangle is right-angled and so we can use <a href="../sine-cosine-tangent.html">sine, cosine and tangent</a> to find how the <b>side</b>, <b>radius</b>, <b>apothem</b> and <b>n</b> (number of sides) are related:</p>
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<div class="simple">
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<table width="580" border="0" align="center" cellpadding="5">
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<tr>
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<td>sin(<font size="+1" face="Times New Roman, Times, serif">π</font>/n) = (Side/2) / Radius</td>
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<td><img src="../images/style/right-arrow.gif" width="46" height="46" alt="right arrow" /></td>
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<td><b>Side = 2 × Radius × sin(<font size="+1" face="Times New Roman, Times, serif">π</font>/n) </b></td>
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</tr>
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<tr>
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<td>cos(<font size="+1" face="Times New Roman, Times, serif">π</font>/n) = Apothem / Radius</td>
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<td><img src="../images/style/right-arrow.gif" width="46" height="46" alt="right arrow" /></td>
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<td><b>Apothem = Radius × cos(<font size="+1" face="Times New Roman, Times, serif">π</font>/n) </b></td>
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</tr>
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<tr>
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<td>tan(<font size="+1" face="Times New Roman, Times, serif">π</font>/n) = (Side/2) / Apothem</td>
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<td><img src="../images/style/right-arrow.gif" width="46" height="46" alt="right arrow" /></td>
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<td><b>Side = 2 × Apothem × tan(<font size="+1" face="Times New Roman, Times, serif">π</font>/n) </b></td>
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</tr>
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</table>
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</div>
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<p>There are a lot more relationships like those (most of them just "re-arrangements"), but those will do for now.</p>
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<h2>More Area Formulas</h2>
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<p>We can use that to calculate the area when we only know the Apothem: </p>
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<div class="center80">
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<div class="tbl">
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<div class="row"><span class="left"><b>Area of Small Triangle</b></span><span class="right">= ½ × Apothem × (Side/2)</span></div>
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</div>
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<p>And we know (from the "tan" formula above) that:</p>
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<p class="center"> <b>Side = 2 × Apothem × tan(<font size="+1" face="Times New Roman, Times, serif">π</font>/n)</b></p>
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<p>So:</p>
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<div class="tbl">
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<div class="row"><span class="left"><b>Area of Small Triangle</b></span><span class="right">= ½ × Apothem × (Apothem × tan(<font size="+1" face="Times New Roman, Times, serif">π</font>/n))</span></div>
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<div class="row"><span class="left"> </span><span class="right"><b>= ½ × Apothem<sup>2</sup> × tan(<font size="+1" face="Times New Roman, Times, serif">π</font>/n)</b></span></div>
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</div>
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</div>
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<p>And there are 2 such triangles per side, or <b>2n</b> for the <b>whole polygon</b>:</p>
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<p class="center larger">Area of Polygon = n × Apothem<sup>2</sup> × tan(<font size="+1" face="Times New Roman, Times, serif">π</font>/n)</p>
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<p>When we don't know the Apothem, we can use the same formula but re-worked for Radius or for Side:</p>
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<p class="center larger">Area of Polygon = ½ × n × Radius<sup>2</sup> × sin(2 × <font size="+1" face="Times New Roman, Times, serif">π</font>/n)</p>
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<p class="center larger">Area of Polygon = ¼ × n × Side<sup>2</sup> / tan(<font size="+1" face="Times New Roman, Times, serif">π</font>/n)</p>
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<h2>A Table of Values</h2>
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<p>And here is a table of Side, Apothem and Area compared to a Radius of "1", using the formulas we have worked out:</p>
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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>Type</th>
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<th>Name when
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<br /> Regular</th>
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<th>Sides
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<br /> (n) </th>
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<th>Shape</th>
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<th>Interior Angle</th>
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<th>Radius</th>
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<th>Side</th>
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<th>Apothem</th>
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<th>Area</th>
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</tr>
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<tr align="center">
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<td><a href="../triangle.html">Triangle</a>
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<br /> <i>(or Trigon)</i></td>
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<th>Equilateral
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<br /> Triangle</th>
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<td><b>3</b></td>
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<td><img src="images/triangle-regular.svg" height="50" alt="regular triangle" /></td>
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<td>60°</td>
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<td>1</td>
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<td>1.732
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<br /> (√3)</td>
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<td>0.5</td>
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<td>1.299
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<br /> (¾√3) </td>
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</tr>
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<tr align="center">
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<td><a href="../quadrilaterals.html">Quadrilateral</a><i><br />
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(or Tetragon)</i></td>
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<th><a href="square.html">Square</a></th>
|
|
<td><b>4</b></td>
|
|
<td><img src="images/quadrilateral-regular.svg" height="50" alt="regular quadrilateral" /></td>
|
|
<td>90°</td>
|
|
<td>1</td>
|
|
<td>1.414
|
|
<br /> (√2) </td>
|
|
<td>0.707
|
|
<br /> (1/√2) </td>
|
|
<td>2</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td><a href="pentagon.html">Pentagon</a></td>
|
|
<th>Regular
|
|
<br /> Pentagon</th>
|
|
<td><b>5</b></td>
|
|
<td><b><img src="images/pentagon-regular.svg" height="50" alt="pentagon regular" /></b></td>
|
|
<td>108°</td>
|
|
<td>1</td>
|
|
<td>1.176</td>
|
|
<td>0.809</td>
|
|
<td>2.378</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td><a href="hexagon.html">Hexagon</a></td>
|
|
<th>Regular
|
|
<br /> Hexagon</th>
|
|
<td><b>6</b></td>
|
|
<td><b><img src="images/hexagon-regular.svg" alt="hexagon regular" height="50" /></b></td>
|
|
<td>120°</td>
|
|
<td>1</td>
|
|
<td>1</td>
|
|
<td>0.866
|
|
<br /> (½√3) </td>
|
|
<td>2.598
|
|
<br /> ((3/2)√3)</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td>Heptagon
|
|
<br /> <i>(or Septagon)</i></td>
|
|
<th>Regular
|
|
<br /> Heptagon</th>
|
|
<td><b>7</b></td>
|
|
<td><b><img src="images/heptagon-regular.svg" alt="heptagon refular" height="50" /></b></td>
|
|
<td>128.571°</td>
|
|
<td>1</td>
|
|
<td>0.868</td>
|
|
<td>0.901</td>
|
|
<td>2.736</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td><a href="octagon.html">Octagon</a></td>
|
|
<th>Regular
|
|
<br /> Octagon</th>
|
|
<td><b>8</b></td>
|
|
<td><b><img src="images/octagon-regular.svg" alt="octagon regular" height="50" /></b></td>
|
|
<td>135°</td>
|
|
<td>1</td>
|
|
<td>0.765</td>
|
|
<td>0.924</td>
|
|
<td>2.828
|
|
<br /> (2√2)</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td>...</td>
|
|
<th>...</th>
|
|
<td> </td>
|
|
<td> </td>
|
|
<td> </td>
|
|
<td> </td>
|
|
<td> </td>
|
|
<td> </td>
|
|
<td> </td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td>Pentacontagon</td>
|
|
<th>Regular
|
|
<br /> Pentacontagon</th>
|
|
<td><b>50</b></td>
|
|
<td> </td>
|
|
<td>172.8°</td>
|
|
<td>1</td>
|
|
<td>0.126
|
|
<br /> </td>
|
|
<td>0.998
|
|
<br /> </td>
|
|
<td>3.133</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td colspan="9">(Note: values correct to 3 decimal places only)</td>
|
|
</tr>
|
|
</table>
|
|
</div>
|
|
|
|
<p> </p>
|
|
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/regular-polygon-graph.svg" alt="regular polygon graph" /></p>
|
|
<h2>Graph</h2>
|
|
<p>And here is a graph of the table above, but with number of sides ("n") from 3 to 30.</p>
|
|
<p>Notice that as "n" gets bigger, the Apothem is tending towards 1 (equal to the Radius) and that the Area is tending towards <b><font size="+1" face="Times New Roman, Times, serif">π</font></b> = 3.14159..., just like a circle.</p>
|
|
<p>What is the Side length tending towards?</p>
|
|
<p> </p>
|
|
<div class="questions">
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<script type="text/javascript">
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</script> </div>
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<div class="related"> <a href="polygons.html">Polygons</a> <a href="polygons-diagonals.html">Diagonals of Polygons</a> <a href="../sine-cosine-tangent.html">Sine, cosine and tangent</a> <a href="index.html">Geometry Index</a> </div>
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