Skip to content

HTTPS clone URL

Subversion checkout URL

You can clone with HTTPS or Subversion.

Download ZIP
Fetching contributors…

Cannot retrieve contributors at this time

110 lines (107 sloc) 5.175 kb
<!DOCTYPE html>
<html data-require="math math-format graphie graphie-geometry graphie-polygon word-problems">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Angles of a polygon</title>
<script data-main="../local-only/main.js" src="../local-only/require.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="render-answer-area-here"></div>
<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" data-forms="integer"><var>ANSWER</var></span> <code>\Large{^\circ}</code>
</div>
<div class="hints">
<p>There are a couple of ways to approach this problem.</p>
<div>
<p>
Since this polygon has <code><var>SIDES</var></code> sides,
we can draw <code><var>SIDES</var></code> 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 <code>360^{\circ}</code> because the circle in the middle is extra.</p>
<p>There are <code>180^{\circ}</code> in a triangle.</p>
<p><code>\begin{align*}&amp;<var>SIDES</var> \times 180^{\circ} - 360^{\circ} \\
&amp;= <var>SIDES * 180</var>^{\circ} - 360^{\circ} \\
&amp;= <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>cardinalThrough20( SIDES )</var> sides to make two 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 data-if="isSingular(SIDES - 4)">
There is one side between the orange triangles, to make one additional triangle.
</p><p data-else="">
There are <code><var>SIDES - 4</var></code> sides between the orange triangles,
to make <code><var>SIDES - 4</var></code> additional triangles.
</p>
<p>
We chopped this polygon into <code><var>SIDES - 2</var></code> triangles,
and each triangle's angles sum to <code>180^{\circ}</code>.
</p>
<div>
<p><code><var>SIDES - 2</var> \times 180^{\circ} = <var>ANSWER</var>^{\circ}</code></p>
<p>The sum of the polygon's interior angles is <code><var>ANSWER</var>^{\circ}</code>.</p>
</div>
</div>
</div>
<div id="exterior">
<p class="question">What is the sum of this polygon's exterior angles?</p>
<div class="render-answer-area-here"></div>
<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" data-forms="integer"><var>360</var></span> <code>\Large{^\circ}</code>
</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.</p>
<p>Therefore, the sum of the exterior angles is <code>360^{\circ}</code>.</p>
</div>
</div>
</div>
</div>
</body>
</html>
Jump to Line
Something went wrong with that request. Please try again.