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<!DOCTYPE html>
<html data-require="math graphie math-format">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Solid geometry</title>
<script src="../khan-exercise.js"></script>
</head>
<body>
<div class="exercise">
<div class="problems">
<div id="cube-V">
<div class="vars">
<var id="s">randRange( 1, 7 )</var>
</div>
<div class="question">
<p>What is the volume of a cube with side length <code><var>s</var></code>?</p>
<div class="graphie">
init({
range: [ [-1, 4], [-0.6, 3.1] ]
});
path([ [0, 0], [0, 2], [2, 2], [2, 0], true ]);
path([ [2, 0], [3, 1], [3, 3], [1, 3], [0, 2] ]);
path([ [2, 2], [3, 3] ]);
label( [1, 0], s, "below" );
</div>
</div>
<p class="solution"><var>s * s * s</var></p>
<div class="hints">
<p>The volume of a cube with side length <code>s</code> is <code>V = s^3</code>.</p>
<p><code>\qquad V = <var>s</var>^3</code></p>
<p><code>\qquad \hphantom{V} = <var>s</var> \cdot <var>s</var> \cdot <var>s</var></code></p>
<p><code>\qquad \hphantom{V} = <var>s * s * s</var></code></p>
</div>
</div>
<div id="cube-T">
<div class="vars">
<var id="s">randRange( 1, 7 )</var>
</div>
<div class="question">
<p>What is the surface area of a cube with side length <code><var>s</var></code>?</p>
<div class="graphie">
init({
range: [ [-1, 4], [-0.6, 3.1] ]
});
path([ [0, 0], [0, 2], [2, 2], [2, 0], true ]);
path([ [2, 0], [3, 1], [3, 3], [1, 3], [0, 2] ]);
path([ [2, 2], [3, 3] ]);
label( [1, 0], s, "below" );
</div>
</div>
<p class="solution"><var>6 * s * s</var></p>
<div class="hints">
<p>The area of each face is simply the area of a square: <code><var>s</var>^2 = <var>s * s</var></code>.</p>
<p>A cube has six faces so the total surface area is <code>6 \cdot <var>s * s</var> = <var>6 * s * s</var></code>.</p>
</div>
</div>
<div id="cylinder-V">
<div class="vars">
<var id="r">randRange( 2, 4 )</var>
<var id="h">randRange( 5, 9 )</var>
</div>
<div class="question">
<p>What is the volume of a cylinder with base radius <code><var>r</var></code> and height <code><var>h</var></code>?</p>
<div class="graphie">
init({
range: [ [-1, 4], [-2, 4] ]
});
arc( [1.5, 3], [1.5, 0.4], 0, 48 );
arc( [1.5, 3], [1.5, 0.4], 70, 360 );
path([ [0, -1], [0, 3] ]);
path([ [3, -1], [3, 3] ]);
arc( [1.5, -1], [1.5, 0.4], 0, 180, {
strokeDasharray: "- "
} );
arc( [1.5, -1], [1.5, 0.4], 180, 360 );
path([ [1.5, 3], [3, 3] ]);
label( [2.25, 3], r, "above" );
label( [3, 1], h, "right" );
</div>
</div>
<div class="solution" data-forms="pi"><var>Math.PI * r * r * h</var></div>
<div class="hints">
<p>The area of the base is simply the area of a circle: <code>\pi r^2 = \pi \cdot <var>r</var>^2 = <var>r * r</var> \pi</code>.</p>
<p>The volume of the cylinder is the area of the base times the height: <code>B \cdot h = <var>r * r</var>\pi \cdot <var>h</var> = <var>r * r * h</var>\pi</code>.</p>
</div>
</div>
<div id="cylinder-T">
<div class="vars">
<var id="r">randRange( 2, 4 )</var>
<var id="h">randRange( 5, 9 )</var>
</div>
<div class="question">
<p>What is the surface area of a cylinder with base radius <code><var>r</var></code> and height <code><var>h</var></code>?</p>
<div class="graphie">
init({
range: [ [-1, 4], [-2, 4] ]
});
arc( [1.5, 3], [1.5, 0.4], 0, 48 );
arc( [1.5, 3], [1.5, 0.4], 70, 360 );
path([ [0, -1], [0, 3] ]);
path([ [3, -1], [3, 3] ]);
arc( [1.5, -1], [1.5, 0.4], 0, 180, {
strokeDasharray: "- "
} );
arc( [1.5, -1], [1.5, 0.4], 180, 360 );
path([ [1.5, 3], [3, 3] ]);
label( [2.25, 3], r, "above" );
label( [3, 1], h, "right" );
</div>
</div>
<div class="solution" data-forms="pi"><var>Math.PI * 2 * r * ( r + h )</var></div>
<div class="hints">
<p>The areas of the top and the base are simply the area of a circle: <code>\pi r^2 = \pi \cdot <var>r</var>^2 = <var>r * r</var> \pi</code>.</p>
<p>The lateral surface area is the same as the area of a rectangle with height <code><var>h</var></code> and width equal to the circumference of the base.</p>
<p>That circumference is <code>2 \pi r = 2\pi \cdot <var>r</var> = <var>2 * r</var>\pi</code>.</p>
<p>Thus, the lateral surface area is <code>wh = <var>2 * r</var> \pi \cdot <var>h</var> = <var>2 * r * h</var> \pi</code>.</p>
<p>The total surface area is <code><var>r * r</var> \pi + <var>r * r</var> \pi + <var>2 * r * h</var> \pi = <var>2 * r * ( r + h )</var>\pi</code>.</p>
</div>
</div>
<div id="box-V">
<div class="vars" data-ensure="d > h">
<var id="w">randRange( 5, 9 )</var>
<var id="h">randRange( 2, 4 )</var>
<var id="d">randRange( 2, 4 )</var>
</div>
<div class="question">
<p>What is the volume of a rectangular box with width <code><var>w</var></code>, height <code><var>h</var></code>, and depth <code><var>d</var></code>?</p>
<div class="graphie">
init({
range: [ [-1, 7], [-1, 3.5] ]
});
var wv = [ 4, 0 ];
var hv = [ 0, 2 ];
var dv = [ 2, 1 ];
path([ [0, 0], hv, [wv[0] + hv[0], wv[1] + hv[1]], wv, true ]);
path([ hv, [hv[0] + dv[0], hv[1] + dv[1]], [wv[0] + hv[0] + dv[0], wv[1] + hv[1] + dv[1]], [wv[0] + dv[0], wv[1] + dv[1]], wv ]);
path([ [wv[0] + hv[0], wv[1] + hv[1]], [wv[0] + hv[0] + dv[0], wv[1] + hv[1] + dv[1]] ]);
label( [0.5 * wv[0], 0.5 * wv[1]], w, "below" );
label( [0.5 * hv[0], 0.5 * hv[1]], h, "left" );
label( [wv[0] + 0.5 * dv[0], wv[1] + 0.5 * dv[1]], d, "above" );
</div>
</div>
<p class="solution"><var>w * h * d</var></p>
<div class="hints">
<p>The volume of a right rectangular prism with width <code>w</code>, height <code>h</code>, and depth <code>d</code> is <code>V = whd</code>.</p>
<p>In this case, the volume is <code><var>w</var> \cdot <var>h</var> \cdot <var>d</var> = <var>w * h * d</var></code>.</p>
</div>
</div>
<div id="box-T">
<div class="vars" data-ensure="d > h">
<var id="w">randRange( 5, 9 )</var>
<var id="h">randRange( 2, 4 )</var>
<var id="d">randRange( 2, 4 )</var>
</div>
<div class="question">
<p>What is the surface area of a rectangular box with width <code><var>w</var></code>, height <code><var>h</var></code>, and depth <code><var>d</var></code>?</p>
<div class="graphie">
init({
range: [ [-1, 7], [-1, 3.5] ]
});
var wv = [ 4, 0 ];
var hv = [ 0, 2 ];
var dv = [ 2, 1 ];
path([ [0, 0], hv, [wv[0] + hv[0], wv[1] + hv[1]], wv, true ]);
path([ hv, [hv[0] + dv[0], hv[1] + dv[1]], [wv[0] + hv[0] + dv[0], wv[1] + hv[1] + dv[1]], [wv[0] + dv[0], wv[1] + dv[1]], wv ]);
path([ [wv[0] + hv[0], wv[1] + hv[1]], [wv[0] + hv[0] + dv[0], wv[1] + hv[1] + dv[1]] ]);
label( [0.5 * wv[0], 0.5 * wv[1]], w, "below" );
label( [0.5 * hv[0], 0.5 * hv[1]], h, "left" );
label( [wv[0] + 0.5 * dv[0], wv[1] + 0.5 * dv[1]], d, "above" );
</div>
</div>
<p class="solution"><var>2 * ( w * h + w * d + h * d )</var></p>
<div class="hints">
<p>The surface area of a right rectangular prism with width <code>w</code>, height <code>h</code>, and depth <code>d</code> is <code>T = 2wh + 2wd + 2hd</code>.</p>
<p>In this case, the surface area is <code>2 \cdot (<var>w</var> \cdot <var>h</var>) + 2 \cdot (<var>w</var> \cdot <var>d</var>) + 2 \cdot (<var>h</var> \cdot <var>d</var>) = <var>2 * ( w * h + w * d + h * d )</var></code>.</p>
</div>
</div>
<div id="tripri-V">
<div class="vars" data-ensure="w * h % 2 === 0 && h < w && w < d">
<var id="w">randRange( 4, 7 )</var>
<var id="h">randRange( 3, 4 )</var>
<var id="d">randRange( 5, 9 )</var>
</div>
<div class="question">
<p>What is the volume of a triangular prism of depth <code><var>d</var></code> whose base has width <code><var>w</var></code> and height <code><var>h</var></code>?</p>
<div class="graphie">
init({
range: [ [-1, 7], [-1, 3.5] ]
});
var wv = [ 3, 0 ];
var hv = [ 2, 2 ];
var dv = [ 3, 1 ];
path([ [0, 0], dv, [hv[0] + dv[0], hv[1] + dv[1]], [wv[0] + dv[0], wv[1] + dv[1]], dv ], {
strokeDasharray: "- ",
stroke: "#ddd"
});
path([ [0, 0], hv, wv, true ]);
path([ hv, [hv[0] + dv[0], hv[1] + dv[1]], [wv[0] + dv[0], wv[1] + dv[1]], wv ]);
// altitude
path([ [hv[0], 0], hv ]);
label( [0.5 * wv[0], 0.5 * wv[1]], w, "below" );
label( [hv[0], 0.5 * hv[1]], h, "left" );
label( [wv[0] + 0.5 * dv[0], wv[1] + 0.5 * dv[1]], d, "below right" );
</div>
</div>
<p class="solution"><var>w * h * d / 2</var></p>
<div class="hints">
<p>The area of the triangular base is <code>\frac12 w h = \frac12 (<var>w</var> \cdot <var>h</var>) = <var>w * h / 2</var></code>.</p>
<p>The volume of the prism is the area of the base times the depth: <code>B \cdot d = <var>w * h / 2</var> \cdot <var>d</var> = <var>w * h * d / 2</var></code>.</p>
</div>
</div>
</div>
</div>
</body>
</html>