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203 lines
8.5 KiB
HTML
203 lines
8.5 KiB
HTML
<!DOCTYPE html>
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<title>Polyhedrons</title>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Polyhedrons</h1>
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<p class="center">A <b>polyhedron</b> is a solid with flat faces<br>
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(from Greek poly- meaning "many" and -hedron meaning "face").</p>
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<div class="center80">
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<p class="center">Each face is a <a href="polygons.html">polygon</a> (a flat shape with straight sides).</p>
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</div>
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<h2>Examples of Polyhedra:</h2>
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<div class="centerfull">
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<div class="boxa"><img src="images/cube.svg" alt="cube" height="139" width="138"><span class="large">Cube</span><br>
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Its faces are all squares</div>
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<div class="boxa"><img src="images/poly-tri-prism.svg" alt="triangular prism" height="70" width="105"><span class="large">Triangular Prism</span><br>
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Its faces are triangles<br>
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and rectangles</div>
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<div class="boxa"><img src="images/poly-dodecahedron.svg" alt="dodecahedron" height="93" width="93"><span class="large">Dodecahedron</span><br>
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What faces does it have?</div>
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</div>
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<p class="center"><b>No curved surfaces</b>: cones, spheres and cylinders are <b>not</b> polyhedrons.</p>
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<h2>Common Polyhedra</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 style="text-align:center;"><a href="hexahedron.html"><img src="images/poly-cube.svg" alt="cube or hexahedron" height="88" width="88"></a> <a href="cuboids-rectangular-prisms.html"><img src="images/poly-square-prism.svg" alt="square prism" height="79" width="104"></a></td>
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<td><a href="hexahedron.html">Cubes</a> and<br>
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<a href="cuboids-rectangular-prisms.html">Cuboids</a> (<a href="../cuboid.html">Volume<br>
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of a Cuboid</a>) </td>
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</tr>
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<tr>
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<td style="text-align:center;"><a href="tetrahedron.html"><img src="images/poly-tetrahedron.svg" alt="tetrahedron" height="94" width="90"></a> <a href="hexahedron.html"><img src="images/poly-cube.svg" alt="cube or hexahedron" height="88" width="88"></a> <a href="octahedron.html"><img src="images/poly-octahedron.svg" alt="octahedron" height="86" width="85"></a> <a href="dodecahedron.html"><img src="images/poly-dodecahedron.svg" alt="dodecahedron" height="93" width="93"></a> <a href="icosahedron.html"><img src="images/poly-icosahedron.svg" alt="icosahedron" height="94" width="81"></a></td>
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<td><a href="../platonic_solids.html">Platonic Solids</a></td>
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</tr>
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<tr>
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<td style="text-align:center;"><a href="prisms.html"><img src="images/poly-tri-prism.svg" alt="triangular prism" height="70" width="105"></a> <a href="prisms.html"><img src="images/poly-square-prism.svg" alt="square prism" height="79" width="104"></a> <a href="prisms.html"><img src="images/poly-pent-prism.svg" alt="pentagonal prism" height="73" width="113"></a></td>
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<td><a href="prisms.html">Prisms</a></td>
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</tr>
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<tr>
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<td style="text-align:center;"><a href="triangular-pyramid.html"><img src="images/poly-tri-pyr.svg" alt="triangular pyramid" height="94" width="100"></a> <a href="square-pyramid.html"><img src="images/poly-square-pyr.svg" alt="square pyramid" height="78" width="115"></a> <a href="pentagonal-pyramid.html"><img src="images/poly-pent-pyr.svg" alt="pentagonal pyramid" height="94" width="90"></a></td>
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<td><a href="pyramids.html">Pyramids</a></td>
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</tr>
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</tbody></table><p class="words">Note: the plural of <b>polyhedron</b> is either <b>polyhedrons</b> or <b>polyhedra</b></p>
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<h2>Many More</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>
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<a href="polyhedron-models.html"><img src="images/polyhedron-models-thumb.gif" alt="Animated Polyhedron Models" height="150" width="150"></a>
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</td>
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<td> </td>
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<td>
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<p>Explore 100s of <a href="polyhedron-models.html">Animated Polyhedron Models</a>.</p>
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<p>You can also see some <a href="polyhedra-images.html">Images of Polyhedra</a> if you want.</p>
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</td>
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</tr>
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</tbody></table>
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<h2>Counting Faces, Vertices and Edges</h2>
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<p>When we count the number of faces (the flat surfaces), vertices (corner points), and edges of a polyhedron we discover an interesting thing:</p>
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<p class="def larger" align="center">The number of <b>faces</b><br>
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<i>plus</i> the number of <b>vertices</b><br>
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<i>minus</i> the number of <b>edges</b> equals <b>2</b></p>
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<p>This can be written neatly as a little equation:</p>
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<h3 align="center">F + V − E = 2</h3>
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<p>It is known as <a href="eulers-formula.html">Euler's Formula</a> (or the "Polyhedral Formula") and is very useful to make sure we have counted correctly!</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/cube.svg" alt="cube" height="139" width="138"></p>
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<h3>Example: Cube</h3>
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<p>A cube has:</p>
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<ul>
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<li>6 Faces</li>
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<li>8 Vertices (corner points)</li>
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<li>12 Edges</li>
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</ul>
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<p class="large center">F + V − E = 6 + 8 − 12 = <b>2</b></p>
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</div>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/poly-tri-prism.svg" alt="triangular prism" height="70" width="105"><br></p>
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<h3>Example: Triangular Prism</h3>
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<p>This prism has:</p>
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<ul>
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<li>5 Faces</li>
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<li>6 Vertices (corner points)</li>
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<li>9 Edges</li>
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</ul>
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<p class="large center">F + V − E = 5 + 6 − 9 = <b>2</b></p>
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</div>
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<p>But there are cases where it does <b>not</b> work! Read <a href="eulers-formula.html">Euler's Formula</a> for more.</p>
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<p> </p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/diagonal.svg" alt="diagonal" height="260" width="263"></p>
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<h2>Diagonals</h2>
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<p>A diagonal is a straight line inside a shape that goes from one corner to another (but not an edge).</p>
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<p>A polyhedron can have lots of diagonals. Can you think of one without diagonals?</p>
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<div style="clear:both"></div>
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<div class="questions">6082, 6083, 6084, 6085, 6086, 6087, 2145, 2146, 2147, 3374, 3375, 3376</div>
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<div class="related">
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<a href="index.html">Geometry</a>
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<a href="solid-geometry.html">Solid Geometry</a>
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<a href="../platonic_solids.html">Platonic Solids</a>
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<a href="prisms.html">Prisms</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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