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171 lines
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HTML
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<title>Pythagoras Theorem in 3D</title>
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<h1>Pythagoras' Theorem in 3D</h1>
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<h2>In 2D</h2>
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<p>First, let us have a quick refresher in two dimensions:</p>
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<p style="float:left; margin: 10px;"><img src="../images/pythagoras.jpg" alt="pythagoras" height="118" width="58"><br>
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<span class="tiny"><i>Pythagoras</i></span></p>
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<p class="larger center"><i> When a triangle has a right angle (90°) ...</i></p>
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<p class="larger center"><i>... and squares are made on each
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of the three sides, ...</i></p>
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<div class="script" style="height: 360px;">
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images/pyth1.js
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</div>
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<p><i>... then the biggest square has the <b>exact same area</b> as the other two squares put together!</i></p><br>
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<div style="clear:both"></div>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/pythagoras-abc.svg" alt="Pythagoras" height="242" width="221"></p>
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<p>It is called "Pythagoras' Theorem" and can be written in one short equation:</p>
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<p class="largest" align="center">a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup></p>
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<p class="center"><img src="images/pythagoras-squares.svg" alt="pythagoras squares a^2 + b^2 = c^2" height="109" width="375"></p>
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<p>Note:</p>
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<ul>
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<li><b>c</b> is the <span class="larger"> <b>longest side</b> of the triangle</span></li>
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<li><b>a</b> and <b>b</b> are the other two sides</li>
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</ul>
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<p>And when we want to know the distance "c" we take the square root:</p>
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<div class="tbl"></div>
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<p class="largest" align="center">c<sup>2</sup> = a<sup>2</sup> + b<sup>2</sup></p>
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<p class="largest" align="center">c = √(a<sup>2</sup> + b<sup>2</sup>)</p>
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<p>You can read more about it at <a href="../pythagoras.html">Pythagoras' Theorem</a>, but here we see how it can be extended into <b>3 Dimensions</b>.</p>
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<h2>In 3D</h2>
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<p>Let's say we want the distance from the bottom-most left front corner to the top-most right back corner of this cuboid:</p>
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<p class="center"><img src="images/pythagoras-3d-a.svg" alt="pythagoras 3d" height="163" width="207"></p>
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<p>First let's just do the triangle on the bottom.</p>
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<p>Pythagoras tells us that <b>c = √(x<sup>2</sup> + y<sup>2</sup>)</b></p>
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<p class="center"><img src="images/pythagoras-3d-b.svg" alt="pythagoras 3d" height="163" width="207"></p>
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<p>Now we make another triangle with its base along the "<b>√(x<sup>2</sup> + y<sup>2</sup>)</b>" side of the previous triangle, and going up to the far corner:</p>
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<p class="center"><img src="images/pythagoras-3d-c.svg" alt="pythagoras 3d" height="163" width="207"></p>
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<p>We can use Pythagoras again, but this time the two sides are <b> √(x<sup>2</sup> + y<sup>2</sup>)</b> and <b> z</b>, and we get this formula:</p>
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<p class="center"><img src="images/pythagoras-3d-xyz.svg" alt="pythagoras 3d" height="79" width="165"></p>
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<p>And the final result is:</p>
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<p class="center"><img src="images/pythagoras-3d-d.svg" alt="pythagoras 3d" height="163" width="207"></p>
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<p> </p>
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<p>So it is all part of a pattern that extends onwards:</p>
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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>
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<th>Dimensions</th>
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<th>Pythagoras</th>
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<th>Distance "c"</th>
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</tr>
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<tr>
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<td style="text-align:center;"><b>1</b></td>
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<td>c<sup>2</sup> = x<sup>2</sup></td>
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<td>√(x<sup>2</sup>) = x</td>
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</tr>
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<tr>
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<td style="text-align:center;"><b>2</b></td>
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<td>c<sup>2</sup> = x<sup>2</sup> + y<sup>2</sup></td>
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<td>√(x<sup>2</sup> + y<sup>2</sup>)</td>
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</tr>
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<tr>
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<td style="text-align:center;"><b>3</b></td>
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<td>c<sup>2</sup> = x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup></td>
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<td>√(x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup>)</td>
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</tr>
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<tr>
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<td style="text-align:center;">...</td>
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<td>...</td>
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<td>...</td>
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</tr>
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<tr>
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<td style="text-align:center;"><b>n</b></td>
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<td>c<sup>2</sup> = a<sub>1</sub><sup>2</sup> + a<sub>2</sub><sup>2</sup> + ... + a<sub>n</sub><sup>2</sup></td>
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<td>√(a<sub>1</sub><sup>2</sup> + a<sub>2</sub><sup>2</sup> + ... + a<sub>n</sub><sup>2</sup>)</td>
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</tr>
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</tbody></table>
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</div>
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<p> </p>
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<p>So next time you need an n-dimensional distance you will know how to calculate it!</p>
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<p> </p>
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<div class="related">
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<a href="../pythagoras.html">Pythagoras' Theorem</a>
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<a href="../right_angle_triangle.html">Right Angled Triangles</a>
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<a href="../activity/fishing-rod.html">The Fishing Rod</a>
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<a href="pythagorean-theorem-proof.html">Pythagorean Theorem Algebra Proof</a>
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