Skip to content
This repository
Fetching contributors…

Cannot retrieve contributors at this time

file 94 lines (92 sloc) 4.472 kb
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94
<!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 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>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</p>
<p>Therefore, the sum of the exterior angles is 360 degrees.</p>
            </div>
        </div>
    </div>
    </div>
</body>
</html>
Something went wrong with that request. Please try again.