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153 lines
7.3 KiB
HTML
153 lines
7.3 KiB
HTML
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<!-- Mirrored from www.mathsisfun.com/geometry/cone-sphere-cylinder.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:37:52 GMT -->
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<meta http-equiv="content-type" content="text/html; charset=utf-8">
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<!-- #BeginEditable "doctitle" -->
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<title>Cone vs Sphere vs Cylinder</title>
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<meta name="description" content="We get this amazing thing that the volume of a cone and sphere together make a cylinder (assuming they fit each other perfectly)">
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<h1 class="center"> Cone vs Sphere vs Cylinder</h1>
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<p class="center"><img src="images/cone-sphere-cylinder.svg" alt="cone sphere cylinder"></p>
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<h2>Volume of a Cone vs Cylinder</h2>
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<p>Let's fit a <a href="cylinder.html">cylinder</a> around a <a href="cone.html">cone</a>.</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/cone-vs-cylinder.svg" alt="cone vs cylinder"></p>
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<p>The volume formulas for cones and cylinders are very similar: </p>
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<div class="simple">
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<table align="center" border="0">
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<tbody>
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<tr valign="middle">
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<td align="right">The volume of a cylinder is: </td>
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<td align="right"><span class="center larger"><span class="times"> π</span> × r<sup>2 </sup>× h</span></td>
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</tr>
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<tr valign="middle">
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<td align="right">The volume of a cone is: </td>
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<td align="right"><span class="center larger"> <span class="intbl">
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<em>1</em>
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<strong>3</strong>
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</span> <span class="times"> π</span> × r<sup>2 </sup>× h</span></td>
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</tr>
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</tbody></table>
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</div><div style="clear:both"></div>
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<p>So the cone's volume is exactly one third ( <span class="intbl">
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<em>1</em>
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<strong>3</strong>
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</span> ) of a cylinder's volume.</p>
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<p><i>(Try to imagine 3 cones fitting inside a cylinder, if you can!)</i></p>
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<h2>Volume of a Sphere vs Cylinder</h2>
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<p>Now let's fit a cylinder around a <a href="sphere.html">sphere</a> .</p>
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<p><span class="large"><b></b></span>We must now make the cylinder's height <span class="large"><b>2r</b></span> so the sphere fits perfectly inside. </p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/cylinder-vs-sphere.svg" alt="cylinder vs sphere"></p>
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<div class="simple">
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<table align="center" border="0">
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<tbody>
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<tr valign="middle">
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<td align="right">The volume of the cylinder is: </td>
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<td align="right"><span class="center larger"><span class="times"> π</span> × r<sup>2 </sup>× h = 2 <span class="times">π</span> × r<sup>3</sup></span></td>
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</tr>
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<tr valign="middle">
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<td align="right">The volume of the sphere is: </td>
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<td align="right"><span class="center larger"> <span class="intbl">
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<em>4</em>
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<strong>3</strong>
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</span> <span class="times"> π</span> × r<sup>3 </sup></span></td>
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</tr>
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</tbody></table>
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</div>
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<p>So the sphere's volume is <span class="center larger"><span class="intbl">
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<em>4</em>
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<strong>3</strong>
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</span></span> vs <span class="center larger">2</span> for the cylinder</p>
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<p>Or more simply the sphere's volume is <span class="center larger"><span class="intbl">
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<em>2</em>
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<strong>3</strong>
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</span></span> of the cylinder's volume!</p>
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<h2>The Result</h2>
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<p>And so we get this amazing thing that the <b>volume of a cone and sphere together make a cylinder</b> (assuming they fit each other perfectly, so <b>h=2r</b>):</p>
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<p class="center"><img src="images/cone-sphere-cylinder-vol.svg" alt="Cone Sphere Cylinder Volumes"></p>
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<p>Isn't mathematics wonderful?</p>
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<p><i>Question: what is the relationship between the volume of a cone and half a sphere (a hemisphere)?</i></p>
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<p> </p>
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<h2>Surface Area</h2>
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<p>What about their surface areas?</p>
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<p class="center"><img src="images/cone-sphere-cylinder-area.svg" alt="cone sphere cylinder area"></p>
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<p><b>No</b>, it does not work for the cone.</p>
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<p>But we do get the same relationship for the sphere and cylinder (<span class="center larger"><span class="intbl"><em>2</em>
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<strong>3</strong></span></span> vs <b>1</b>)</p>
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<p> </p>
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<p>And there is another interesting thing: if we <b>remove the two ends</b> of the cylinder then its surface area is exactly the same as the sphere:</p>
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<p class="center"><img src="images/sphere-cylinder-area.svg" alt="sphere vs cylinder area"></p>
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<p>Which means that we could reshape a cylinder (of height <b>2r</b> and without its ends) to fit perfectly on a sphere (of radius <b>r</b>):</p>
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<p class="center"><img src="images/sphere-cylinder-area2.svg" alt="sphere vs cylinder area"><br>
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Same Area</p>
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<p>(Research "Archimedes' Hat-Box Theorem" to learn more.)</p>
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<p> </p>
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<div class="related">
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<a href="cone.html">Cone</a>
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<a href="sphere.html">Sphere</a>
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<a href="cylinder.html">Cylinder</a>
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<a href="index.html">Geometry Index</a></div>
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