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<!DOCTYPE html>
<html data-require="math math-format graphie graphie-geometry graphie-polygon word-problems">
<head>
<meta charset="UTF-8" />
<title>Angles of a polygon</title>
<script src="../khan-exercise.js"></script>
</head>
<body>
<div class="exercise">
<div class="vars">
<var id="SIDES">randRange( 5, 7 )</var>
</div>
<div class="problems">
<div id="interior">
<div class="vars">
<var id="ANSWER">180 * ( SIDES - 2 )</var>
<var id="CLONE">{}</var>
</div>
<p class="question">What is the sum of this polygon's interior angles?</p>
<div class="graphie" id="circle">
init({
range: [ [ -5, 5 ], [ -1, 5 ] ],
scale: [ 40, 40 ]
});
graph.polygon = new Polygon( SIDES );
graph.polygon.draw();
CLONE = graph.polygon.clone();
</div>
<div class="solution" data-type="multiple">
<span class="sol"><var>ANSWER</var></span> degrees
</div>
<div class="hints">
<p>There are a couple ways to approach this problem.</p>
<p>Does it help to remember that there are 180 degrees in a triangle?</p>
<div>
<p>Since this polygon has <var>SIDES</var> sides, we can draw <var>SIDES</var> triangles that all meet in the center.</p>
<div class="graphie" data-update="circle">
graph.polygon.drawRadialDiagonals();
</div>
</div>
<p>We can combine all the triangles' angles, and then we must subtract 360 degrees because the circle in the middle is extra.</p>
<p><code>\begin{align*}&<var>SIDES</var> \times 180^{\circ} - 360^{\circ} \\
&= <var>SIDES * 180</var>^{\circ} - 360^{\circ} \\
&= <var>ANSWER</var>^{\circ}\end{align*}</code></p>
<div>
<p>An alternative approach is shown below.</p>
<p>We can use four of the <var>cardinal( SIDES )</var> sides to make 2 triangles, as shown in orange.</p>
<div class="graphie" data-id="other">
init({
range: [ [ -5, 5 ], [ -1, 5 ] ]
});
graph.polygon = CLONE;
graph.polygon.draw();
graph.polygon.drawDiagonals( randRange( 0, SIDES - 1 ) );
</div>
</div>
<p>There <var>plural( "is", SIDES - 4 )</var> <var>plural( SIDES - 4, "side" )</var> between the orange triangles, to make <var>SIDES - 4</var> additional <var>plural( "triangle", SIDES - 4 )</var>.</p>
<p>We chopped this polygon into <var>SIDES - 2</var> triangles, and each triangle's angles sum to 180 degrees.</p>
<p><code><var>SIDES - 2</var> \times 180^{\circ} = <var>ANSWER</var>^{\circ}</code></p>
<p>Again, we have found that the sum of the polygon's interior angles is <var>ANSWER</var> degrees.</p>
</div>
</div>
<div id="exterior">
<p class="question">What is the sum of this polygon's exterior angles?</p>
<div class="graphie" id="polygon">
init({
range: [ [ -6, 6 ], [ -2, 7 ] ]
});
graph.polygon = new Polygon( SIDES );
graph.polygon.draw();
</div>
<div class="solution" data-type="multiple">
<span class="sol"><var>360</var></span> degrees
</div>
<div class="hints">
<div>
<p>The exterior angles are shown above.</p>
<div class="graphie" data-update="polygon">
graph.polygon.drawExteriorAngles();
</div>
</div>
<div class="graphie" data-update="polygon">
graph.polygon.animateExteriorAngles( randRange( 0, SIDES - 1 ) );
</div>
<p>The exterior angles fit together to form a circle, summing to 360 degrees.</p>
</div>
</div>
</div>
</div>
</body>
</html>
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