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<title>Interior Angles of Polygons</title>
<meta name="description" content="An Interior Angle is an angle inside a shape. Example: ...">
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<h1 class="center">Interior Angles of Polygons</h1>
<h3 align="center">An Interior Angle is an angle inside a shape</h3>
<p class="center"><img src="images/interior-exterior-angles.svg" alt="interior exterior angles"></p>
<p>Another example:</p>
<p class="center"><img src="images/interior-exterior-angles2.svg" alt="interior exterior angles"></p>
<h2>Triangles</h2>
<p class="def large" align="center">The Interior Angles of a Triangle add up to 180°</p>
<div class="centerfull">
<div class="boxa">
<p>Let's try a triangle:<br>
<img src="images/interior-angles-triangle1.gif" alt="interior angles triangle 90 60 30" height="146" width="246"><span class="larger"><br>
90° + 60° + 30° = 180°</span></p>
<p>It works for this triangle</p>
</div>
<div class="boxa">
<p><br>
Now tilt a line by 10°:<br>
<img src="images/interior-angles-triangle2.gif" alt="interior angles triangle 80 70 30" height="132" width="245"><span class="larger"><br>
80° + 70° + 30° = 180°</span></p>
<p>It still works!<br>
One angle went <b>up</b> by 10°,<br>
and the other went <b>down</b> by 10°</p>
</div></div>
<h2>Quadrilaterals (Squares, etc)</h2>
<p>(A Quadrilateral has 4 straight sides)</p>
<div class="centerfull">
<div class="boxa">
<p>Let's try a square:<br>
<img src="images/interior-angles-square1.gif" alt="interior angles square 90 90 90 90" height="136" width="135"><span class="larger"><br>
90° + 90° + 90° + 90° = 360°</span></p>
<p>A Square adds up to 360°</p>
</div>
<div class="boxa">
<p><br>
Now tilt a line by 10°:<br>
<img src="images/interior-angles-square2.gif" alt="interior angles 100 90 90 80" height="136" width="134"><span class="larger"><br>
80° + 100° + 90° + 90° = 360°</span></p>
<p>It still adds up to 360°</p>
</div>
</div>
<p class="def large" align="center">The Interior Angles of a Quadrilateral add up to 360°</p>
<h3>Because there are 2 triangles in a square ...</h3>
<p class="center"><img src="images/interior-angles-square3.gif" alt="interior angles 90 (45,45) 90 (45,45)" height="163" width="341"></p>
<p>The interior angles in a triangle add up to <b>180°</b> ...</p>
<p class="center">... and for the square they add up to <b>360° </b>...</p>
<p style="float:right; margin: 0 0 5px 10px;">... because the square can be made from two triangles!</p>
<div style="clear:both"></div>
<h2>Pentagon</h2>
<p style="float:left; margin: 0 10px 5px 0;"><img src="images/interior-angles-pentagon.gif" alt="interior angles pentagon" height="192" width="201"></p>
<p>A pentagon has 5 sides, and can be made from <b>three triangles</b>, so you know what ...</p>
<p class="center">... its interior angles add up to 3 × 180° =<b> 540° </b></p>
<p>And when it is <b>regular</b> (all angles the same), then each angle is 540<b>°</b> / 5 = 108<b>°</b></p>
<p><i>(Exercise: make sure each triangle here adds up to 180°, and check that the pentagon's interior angles add up to 540°)</i></p>
<p class="def large" align="center">The Interior Angles of a Pentagon add up to 540°</p>
<h2>The General Rule</h2>
<p>Each time we add a side (triangle to quadrilateral, quadrilateral to pentagon, etc), we <b>add another 180°</b> to the total: </p>
<div class="simple">
<table align="center" width="90%" border="0">
<tbody>
<tr>
<td style="background-color:rgba(255, 255, 255, 0.8);" width="19%">&nbsp;</td>
<td style="background-color:rgba(255, 255, 255, 0.8);" width="11%">&nbsp;</td>
<td style="background-color:rgba(255, 255, 255, 0.8);" width="30%">&nbsp;</td>
<td style="" colspan="2" align="center">If it is a <b><a href="regular-polygons.html">Regular Polygon</a></b> (all sides are equal, all angles are equal)</td>
</tr>
<tr align="center">
<th width="19%">Shape</th>
<th width="11%">Sides</th>
<th width="30%">Sum of<br>
Interior Angles</th>
<th width="10%">Shape</th>
<th width="30%">Each Angle</th>
</tr>
<tr align="center">
<td width="19%"><a href="../triangle.html">Triangle</a></td>
<td width="11%">3</td>
<td width="30%">180<b>°</b></td>
<td><img src="images/triangle-regular.svg" alt="regular triangle" height="50"></td>
<td width="30%">60<b>°</b></td>
</tr>
<tr align="center">
<td width="19%"><a href="../quadrilaterals.html">Quadrilateral</a></td>
<td width="11%">4</td>
<td width="30%">360<b>°</b></td>
<td><img src="images/quadrilateral-regular.svg" alt="regular quadrilateral" height="50"></td>
<td width="30%">90<b>°</b></td>
</tr>
<tr align="center">
<td width="19%"><a href="pentagon.html">Pentagon</a></td>
<td width="11%">5</td>
<td width="30%">540<b>°</b></td>
<td><b><img src="images/pentagon-regular.svg" alt="pentagon regular" height="50"></b></td>
<td width="30%">108<b>°</b></td>
</tr>
<tr align="center">
<td width="19%"><a href="hexagon.html">Hexagon</a></td>
<td width="11%">6</td>
<td width="30%">720<b>°</b></td>
<td><b><img src="images/hexagon-regular.svg" alt="hexagon regular" height="50"></b></td>
<td width="30%">120<b>°</b></td>
</tr>
<tr align="center">
<td>Heptagon <i>(or Septagon)</i></td>
<td>7</td>
<td>900<b>°</b></td>
<td><b><img src="images/heptagon-regular.svg" alt="heptagon refular" height="50"></b></td>
<td>128.57...<b>°</b></td>
</tr>
<tr align="center">
<td>Octagon</td>
<td>8</td>
<td>1080<b>°</b></td>
<td><b><img src="images/octagon-regular.svg" alt="octagon regular" height="50"></b></td>
<td>135<b>°</b></td>
</tr>
<tr align="center">
<td>Nonagon</td>
<td>9</td>
<td>1260<b>°</b></td>
<td><img src="images/nonagon-regular.svg" alt="nonagon regular" height="50"></td>
<td>140<b>°</b></td>
</tr>
<tr align="center">
<td width="19%">...</td>
<td width="11%">...</td>
<td width="30%">..</td>
<td width="10%">...</td>
<td width="30%">...</td>
</tr>
<tr align="center">
<td class="larger" width="19%">Any Polygon</td>
<td width="11%"><b class="larger">n</b></td>
<td class="larger" width="30%">(<b>n</b>−2) × 180<b>°</b></td>
<td width="10%"><img src="images/n-gon-regular.svg" alt="regular n gon" height="50"></td>
<td class="larger" width="30%">(<b>n</b>−2) × 180<b>°</b> / <b>n</b></td>
</tr>
</tbody></table>
</div>
<p>So the general rule is:</p>
<div class="def">
<p class="larger">Sum of Interior Angles = (<b>n</b>−2) × 180<b>°</b></p>
<p class="larger">Each Angle (of a Regular Polygon) = (<b>n</b>−2) × 180<b>°</b> / <b>n</b></p>
</div>
<p>Perhaps an example will help:</p>
<div class="example">
<h3>Example: What about a Regular Decagon (10 sides) ?</h3>
<p style="float:left; margin: 0 10px 5px 0;"><img src="images/decagon-regular.svg" alt="regular decagon" height="100"></p>
<div class="tbl">
<div class="row"><span class="left">Sum of Interior Angles </span><span class="right">= (<b>n</b>−2) × 180<b>°</b></span></div>
<div class="row"><span class="left">&nbsp;</span><span class="right">= (<b>10</b>−2) × 180<b>°</b></span></div>
<div class="row"><span class="left">&nbsp;</span><span class="right">= 8 × 180°</span></div>
<div class="row"><span class="left">&nbsp;</span><span class="right">= <b>1440°</b></span></div>
</div>
<p>And for a Regular Decagon:</p>
<p class="center">Each interior angle = 1440<b>°</b>/10 = <b>144°</b></p>
</div>
<p>&nbsp;</p>
<div class="words">
<p>Note: Interior Angles are sometimes called "Internal Angles"</p>
</div>
<div class="questions">
<script>getQ(825, 826, 3314, 3315, 827, 828, 1779, 829, 1780, 3316);</script>&nbsp;
</div>
<div class="related">
<a href="interior-angles.html">Interior Angles</a>
<a href="exterior-angles.html">Exterior Angles</a>
<a href="degrees.html">Degrees (Angle)</a>
<a href="../shape.html">2D Shapes</a>
<a href="../triangle.html">Triangles</a>
<a href="../quadrilaterals.html">Quadrilaterals</a>
<a href="index.html">Geometry Index</a>
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