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226 lines
9.4 KiB
HTML
226 lines
9.4 KiB
HTML
<!DOCTYPE html>
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<html lang="en"><!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
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<!-- Mirrored from www.mathsisfun.com/geometry/torus.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:44:32 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>Torus</title>
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<script language="JavaScript" type="text/javascript">reSpell=[["tire","tyre"]];</script>
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<style type="text/css">
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.boxa {
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}
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</style>
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<meta name="keywords" content="math, maths, mathematics, school, homework, education">
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<body id="bodybg">
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<div id="stt"></div>
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<div id="adTop"></div>
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<header>
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<div id="hdr"></div>
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<div id="tran"></div>
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<div id="adHide"></div>
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<div id="cookOK"></div>
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</header>
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<div class="mid">
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<nav>
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<div id="menuWide" class="menu"></div>
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<div id="logo"><a href="../index.html"><img src="../images/style/logo.svg" alt="Math is Fun"></a></div>
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<div id="search" role="search"></div>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Torus</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">Torus Facts</div>
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<p><b>Notice these interesting things:</b></p>
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<p class="center"><img src="images/torus.svg" alt="Torus Radii" height="251" width="333"></p>
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<ul>
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<li>It can be made by revolving a<br>
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small circle (radius <b>r</b>) along a line made<br>
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by a bigger circle (radius <b>R</b>).</li>
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<li>It has no edges or vertices</li>
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<li>It is <b>not</b> a <a href="polyhedron.html">polyhedron</a></li>
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</ul>
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</div>
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<div class="boxb"> <img src="images/torus.jpg" alt="torus in the sky" height="329" width="317"><i><br>
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Torus in the Sky</i>.<br>
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The Torus is such a beautiful solid,<br>
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this one would be fun at the beach ! </div>
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</div>
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<div class="script" style="height: 460px;">
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images/poly-gl.js?mode=torus
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</div>
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<h2>Surface Area</h2>
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<a id="Area"></a>
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<p class="center"><img src="images/torus.svg" alt="Torus Radii" height="251" width="333"></p>
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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><span class="center large">Surface Area </span></td>
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<td><span class="center large">= (2<span class="times">π</span>R) × (2<span class="times">π</span>r)</span></td>
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</tr>
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<tr>
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<td> </td>
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<td><span class="center large">= 4 × <span class="times">π</span><sup>2</sup> × R × r</span></td>
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</tr>
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</tbody></table>
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<div class="example">
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<h3>Example: r = 3 and R = 7</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"><span class="center">= 4 × <span class="times">π</span><sup>2</sup> × R × r</span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="center">= 4 × <span class="times">π</span><sup>2</sup> × 7 × 3</span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="center">= 4 × <span class="times">π</span><sup>2</sup> × 21</span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="center">= 84 × <span class="times">π</span><sup>2</sup></span></span></div>
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<div class="row"><span class="left"> </span><span class="right">≈ 829</span></div>
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</div>
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</div>
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<p>The formula is often written in this shorter way:</p>
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<p class="center large">Surface Area = 4<span class="times">π</span><sup>2</sup>Rr</p>
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<a id="Volume"></a>
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<h2>Volume</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><span class="center large">Volume <br>
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</span></td>
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<td><span class="center large">= (2<span class="times">π</span>R) × (<span class="times">π</span></span><span class="center large"><span class="center large">r</span><span class="center large"><span class="times"></span><sup>2</sup></span>)</span></td>
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</tr>
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<tr>
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<td> </td>
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<td><span class="center large">= 2 × <span class="times">π</span><sup>2</sup> × R × r</span><span class="center large"><span class="times"></span><sup>2</sup></span></td>
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</tr>
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</tbody></table>
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<div class="example">
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<h3>Example: r = 3 and R = 7</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">= 2 × <span class="times">π</span><sup>2</sup> × R × r<sup>2</sup></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><sup>2</sup> × 7 × 3<sup>2</sup></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><sup>2</sup> × 7 × 9</span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="center">= 126 <span class="times">π</span><sup>2</sup></span></span></div>
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<div class="row"><span class="left"> </span><span class="right">≈ 1244</span></div>
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</div>
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</div>
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<p>The formula is often written in this shorter way:</p>
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<p class="center large">Volume = 2<span class="times">π</span><sup>2</sup> Rr<sup>2</sup></p>
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<p> </p>
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<p class="center"><i>Note: Area and volume formulas only work when the torus has a hole!</i></p>
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<h2>Like a Cylinder</h2>
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<p><b>Volume</b>: the volume is the same as if we "unfolded" a torus into a cylinder (of length 2<font size="+1"><font face="Times New Roman, Times, serif"><i>π</i></font></font>R):</p>
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<p class="center"><img src="images/torus-to-cylinder.svg" alt="Torus to Cylinder" style="max-" height="503" width="835"></p>
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<p>As we unfold it, what gets lost from the outer part of the torus is perfectly balanced by what gets gained in the inner part.</p>
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<p><br></p>
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<p><b>Surface Area</b>: the same is true for the surface area, not including the cylinder's bases.</p>
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<p class="center"><img src="images/torus-to-cylinder-area.svg" alt="Torus to Cylinder Area" style="max-" height="250" width="857"></p>
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<p> </p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/torus-cushion.jpg" alt="Torus Cushion Illustration" height="146" width="171"></p>
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And did you know that <i>Torus</i> was the
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Latin word for a <b>cushion</b>?
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<p>(This is not a <i>real</i> roman cushion, just an illustration I made)</p>
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<p>The Volume and Area calculations will not work with this cushion because there is no hole.</p>
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<div style="clear:both"></div>
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<div class="words">
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<p>When we have more than one torus they are called <b>tori</b></p> </div>
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<h2>More Torus Images</h2>
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As the small radius (<b>r</b>) gets larger and larger, the torus goes from looking like a <i><b>Tire</b></i> to a <i><b>Donut:</b></i>
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<p class="center"><img src="images/torus-brass.jpg" alt="Torus Tire" height="153" width="146"><img src="images/torus-brass2a.jpg" alt="Torus Donut" height="153" width="169"></p>
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<p> </p>
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<div class="questions">3391, 3392, 3393, 3394, 3395, 3396, 3397, 3398, 7656, 7657</div>
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<div class="related">
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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="cone.html">Cone</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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