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281 lines
12 KiB
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Cylinder</h1>
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<p class="center"><i>Go to <a href="#Area">Surface Area</a> or <a href="#Volume">Volume</a>.</i></p>
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<div class="centerfull">
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<div class="boxa">
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<div class="head">Cylinder Facts</div>
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<p><b>Notice these interesting things:</b></p>
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<p class="center"><img src="images/cylinder-shiny.jpg" alt="cylinder shiny" height="157" width="180"></p>
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<ul>
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<li>It has a flat base and a<br>
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flat top</li>
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<li>The base is the same as the top</li>
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<li>From base to top the shape stays the same</li>
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<li>It has one curved side</li>
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<li>It is <b>not</b> a <a href="polyhedron.html">polyhedron</a> as it
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has a curved surface</li>
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</ul>
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</div>
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<div class="boxb">
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<div class="script" style="height: 460px;">
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images/poly-gl.js?mode=cylinder
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</div>
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</div>
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</div>
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<p class="center"><img src="images/cylinder-wood.jpg" alt="wooden cylinder" height="197" width="300"><br>
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A wooden cylinder</p>
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<div class="words">
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<p>An object shaped like a cylinder is said to be <b>cylindrical.</b></p>
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</div>
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<h2>Oblique Cylinder</h2>
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<p>When the two ends are directly aligned on each other it is a Right Cylinder otherwise it is an Oblique Cylinder:</p>
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<p class="center"><img src="images/cylinder-oblique.svg" alt="oblique cylinder" height="223" width="292"></p>
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<h2><a id="Area"></a>Surface Area of a Cylinder</h2>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/cylinder-dimensions.svg" alt="Cylinder Dimensions: r=radius, h=height" height="186" width="215"></p>
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<p>The Surface Area has these parts:</p>
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<ul>
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<li><i>Surface Area of Both Ends</i> = 2 × <span class="times">π</span> × r<sup>2</sup></li>
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<li><i>Surface Area of Side</i> = 2 × <span class="times">π</span> × r × h</li>
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</ul>
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<p>Which together make:</p>
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<p class="center large">Surface Area = 2 × <span class="times">π</span> × r × (r+h)</p>
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<div style="clear:both"></div>
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<div class="example">
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<h3>Example: h = 7 and r = 2</h3>
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<div class="tbl">
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<div class="row"><span class="left"><span class="center">Surface Area </span></span><span class="right">= 2 × <span class="times">π</span> × r × (r+h)</span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="center">= 2 × <span class="times">π</span> × 2 × (2+7)</span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="center">= 2 × <span class="times">π</span> × 2 × 9</span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="center">= 36 <span class="times">π</span></span></span></div>
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<div class="row"><span class="left"> </span><span class="right">≈ <b>113.097</b></span></div>
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</div></div>
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<p>Try it yourself: cut some paper so it fits around a cylinder, then unwrap and measure it.</p>
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<p>It will be <b>h</b> high and <b>2<span class="times">π</span>r</b> (the circumference of a circle) long:</p>
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<p class="center"><img src="images/cylinder-area.jpg" alt="cylinder area" height="135" width="360"><br>
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<span class="larger">Side area = <b>2</b></span><span class="times"><b>π</b></span><span class="larger"><b>rh</b></span></p>
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<p>Don't forget the two end bits:</p>
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<p class="center"><img src="images/cylinder-area2.jpg" alt="cylinder area" height="177" width="280"><span class="larger"><br>
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Total
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Surface Area<br>
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= 2(<span class="times">π</span>r<sup>2</sup>) + 2<span class="times">π</span>rh<br>
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= 2<span class="times">π</span>r(r+h)</span></p>
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<a id="Volume"></a>
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<h2>Volume of a Cylinder</h2>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/cylinder-dimensions.svg" alt="Cylinder Dimensions: r=radius, h=height" height="186" width="215"></p>
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<p>To calculate the volume we multiply the area of the base by the height of the cylinder:</p>
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<ul>
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<li>Area of the base: <span class="larger"><span class="times">π</span> × r<sup>2</sup></span></li>
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<li>Height: <span class="larger">h</span></li></ul>
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<p>And we get:</p>
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<p class="center large">Volume = <span class="times">π</span> × r<sup>2 </sup>× h</p>
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<div class="example">
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<h3>Example: h = 7 and r = 2</h3>
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<div class="tbl">
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<div class="row"><span class="left"><span class="center">Volume</span></span><span class="right"><span class="center">= <span class="times">π</span> × r<sup>2</sup> <span class="center larger">× h</span></span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="center">= <span class="times">π</span> × 2<sup>2</sup> <span class="center larger">× 7</span></span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="center">= 28 <span class="times">π</span></span></span></div>
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<div class="row"><span class="left"> </span><span class="right">≈<b> 87.96</b></span></div>
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</div>
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</div>
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<div class="simple"></div>
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<div class="fun">
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<h3>How to remember: <b>Volume = pizza</b></h3>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/cylinder-pizza.svg" alt="Cylinder Pizza" height="85" width="197"></p>
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<p>Imagine you just cooked a <b>pizza</b>.</p>
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<p>The radius is "z", and the thickness "a" is the same everywhere ... what is the volume?</p>
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<p class="center"><b>Volume = </b> <span class="large"><b>pi</b> × <b>z</b> × <b>z</b> × <b>a</b></span></p>
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<p class="center">(we would normally write "pi" as <span class="times">π</span>, and <b>z</b> × <b>z</b> as <b>z<sup>2</sup></b>, but you get the idea!)</p>
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</div>
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<p>Play with it here. The formula also works when it "leans over" (<i>oblique</i>) but remember that the height is always at right angles to the base:</p>
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<iframe src="icylinder.html" scrolling="no" style="width:362px; height:302px; overflow:hidden; margin:auto; display:block; border: none; text-align:center;"></iframe>
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<p>And this is why:</p>
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<p class="center"><img src="images/cylinder-stack-right.jpg" alt="tokens stacked right" height="173" width="139"> <img src="images/cylinder-stack-oblique.jpg" alt="tokens stacked oblique" height="173" width="182"><br>
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The stack leans over, but still has the same volume</p>
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<h2>Volume of a Tank</h2>
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<p>Learn how to find the <a href="cylinder-horizontal-volume.html">volume of a partly filled horizontal cylinder</a> like this one:</p>
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<p class="center"><img src="images/cylinder-horiz.svg" alt="Horizontal cylinder" height="129" width="287"></p>
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<h2>Volume of a Cone vs Cylinder</h2>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/cone-vs-cylinder.svg" alt="cone vs cylinder" height="126" width="232"></p>
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<p> </p>
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<p>The volume formulas for cylinders and <a href="cone.html">cones</a> are very similar:</p>
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<div style="clear:both"></div>
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<div class="simple">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr valign="middle">
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<td style="text-align:right;">Volume of a cylinder: </td>
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<td style="text-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 style="text-align:right;">Volume of a cone: </td>
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<td style="text-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>
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<p>So a 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>In future, order your ice creams in cylinders, not cones, you get 3 times as much!</p>
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<h2>Like a Prism</h2>
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<p>A cylinder is like a prism with an infinite number of sides, see <a href="prism-vs-cylinder.html">Prism vs Cylinder</a>.</p>
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<h2>It Doesn't Have to Be Circular</h2>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/cylinder-elliptical.jpg" alt="wooden elliptical cylinder" height="89" width="221"></p>
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<p>Usually when we say <i>Cylinder</i> we mean a <i>Circular</i> Cylinder, but you can also have <b>Elliptical Cylinders</b>, like this one:<br>
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<br>
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And we can have stranger cylinders!</p>
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<p>So long as the cross-section is curved and is the same from one end to the other, then it is a cylinder. But the area and volume calculations will be different than shown above.</p>
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<h2>More Cylinders</h2>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td><img src="images/cylinder-wood1.jpg" alt="wooden cylinder tall" height="220" width="139"><img src="images/cylinder-wood2.jpg" alt="wooden cylinder short" height="220" width="206"><img src="images/cylinder-wood3.jpg" alt="wooden cylinder thin" height="220" width="101"></td>
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</tr>
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</tbody></table>
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<p class="center"><img src="images/soup-can.svg" alt="soup can" height="239" width="158"><br>
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<i>"We eat what we can<br>
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And what we can't we can"</i></p>
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<p> </p>
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<div class="questions">5495, 5589, 5591, 5592, 869, 870, 3388, 3390, 3387, 3389</div>
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<div class="related">
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<a href="prism-vs-cylinder.html">Prism vs Cylinder</a>
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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="cone-sphere-cylinder.html">Cone vs Sphere vs Cylinder</a>
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<a href="torus.html">Torus</a>
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<a href="prisms.html">Prisms</a>
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<a href="polyhedron.html">Polyhedron</a>
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<a href="../platonic_solids.html">Platonic Solids</a>
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</div>
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<!-- #EndEditable -->
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</article>
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<footer id="footer" class="centerfull noprint"></footer>
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<div id="copyrt">Copyright © 2022 Rod Pierce</div>
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