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168 lines
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HTML
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<title>Pythagoras' Theorem and Areas</title>
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<meta name="description" content="Learn the generalized form of the Pythagorean Theorem">
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1>Pythagoras' Theorem and Areas</h1>
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<h2>Pythagoras' Theorem</h2>
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<p>Let's start with a quick refresher of the famous Pythagoras' Theorem.</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/triangle-abc.svg" alt="triangle abc" style="width: 201px; height: 115px;"></p>
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<p style="text-align: center;">Pythagoras' Theorem says that, in a
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right angled triangle:<br>
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the square of the hypotenuse (<strong>c</strong>) is equal to the sum of
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the squares of the other two sides (<strong>a</strong> and <strong>b</strong>).</p>
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<p class="center large">a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup></p>
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<p>That means we can draw squares on each side:</p>
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<p class="center"><img src="images/pythagoras-squares-area.svg" alt="Pythagoras Areas" class="center"></p>
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<p>And this will be true:</p>
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<p class="largest center">A + B = C</p>
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<p>You can learn more about the <a href="../pythagoras.html">Pythagorean
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Theorem</a> and review its <a href="pythagorean-theorem-proof.html">algebraic
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proof</a>.</p>
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<h2>A More Powerful Pythagorean Theorem </h2>
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<p>Say we want to draw semicircles on each side of a right triangle:</p>
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<div style="text-align: center;"><img src="images/pythagoras-circle.svg" alt="Pythagoras semicircle"><br>
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<strong>A</strong>, <strong>B</strong> and<strong> C</strong>
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are the areas of each<br>
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semicircle with diameters <strong>a</strong>,
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<strong>b</strong> and <strong>c</strong>.</div>
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<p>Maybe A + B = C ?</p>
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<p>But they aren't squares! Yet let's go ahead anyway to see where it leads us.</p>
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<p>OK, the area of a <a href="circle.html">circle</a> with diameter "D" is:</p>
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<p class="center larger"><strong>Area of Circle</strong> = <span class="intbl"><em>1</em> <strong>4</strong> </span> <span class="times">π</span> D<sup>2</sup></p>
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<p>So the area of a semicircle is <b>half</b> of that:</p>
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<p class="center larger"><strong>Area of Semicircle</strong> = <span class="intbl"><em>1</em> <strong>8</strong> </span> <span class="times">π</span> D<sup>2</sup></p>
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<p>And so the area of each semicircle is:</p>
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<p class="center larger"><strong>A</strong> = <span class="intbl"><em>1</em>
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<strong>8</strong> </span> <span class="times">π</span>a<sup>2</sup></p>
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<p class="center larger"><strong>B</strong> = <span class="intbl"><em>1</em>
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<strong>8</strong> </span> <span class="times">π</span>b<sup>2</sup></p>
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<p class="center larger"><strong>C</strong> = <span class="intbl"><em>1</em>
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<strong>8</strong> </span> <span class="times">π</span>c<sup>2</sup></p>
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<p>Now our question:</p>
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<div class="fun">
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<p class="center larger">Does A + B = C ?</p>
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</div>
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<p>Let's substitute the values:</p>
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<p class="center larger">Does <span class="intbl"><em>1</em> <strong>8</strong>
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</span><span class="times">π</span>a<sup>2 </sup>+ <span class="intbl"><em>1</em>
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<strong>8</strong> </span><span class="times">π</span>b<sup>2</sup>
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= <span class="intbl"><em>1</em> <strong>8</strong> </span><span class="times">π</span>c<sup>2</sup> ?</p>
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<p>We can <a href="../algebra/factoring.html">factor out</a> <span class="intbl"><em>1</em>
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<strong>8</strong> </span><span class="times">π</span> and we get:</p>
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<p class="center larger">a<sup>2 </sup>+ b<sup>2</sup> = c<sup>2</sup></p>
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<p>Yes! It is simply Pythagoras' Theorem.</p>
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<p>Therefore, we have shown that Pythagoras' Theorem is true for
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semicircles.</p>
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<p>Will it work for any other shape?</p>
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<div style="text-align: center;"><img src="images/pythagoras-star.svg" alt="Pythagoras' Star"></div>
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<p>Yes! The Pythagorean Theorem can be taken further into a
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shape-generalized form as long as the shapes are <a href="similar.html">similar</a> (has a special meaning in Geometry).</p>
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<p><br></p>
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<div class="def">
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<p class="larger"><b> Shape-Generalization Form of the Pythagorean
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Theorem:<br>
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</b><br>
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Given a right triangle, we can draw <b>similar</b> shapes on
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each side so that the area of the shape constructed on the
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hypotenuse is the sum of the areas of similar shapes constructed on
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the legs of the triangle.</p>
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<p class="largest center">A + B = C</p>
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<p class="larger">Where:</p>
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<ul style="text-align: left;">
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<li><strong>A</strong> is the area of the shape on the
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hypotenuse.<strong></strong></li>
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<li><strong>B</strong> and <strong>C</strong> are the areas of the
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shapes on the legs.</li>
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</ul>
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</div>
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<p><br></p>
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<p>The Theorem still holds for cool shapes that are not polygons, such as
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this amazing dragon!</p>
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<div style="text-align: center;"><img src="images/pythagoras-dragon.svg" alt="Pythagoras' Dragon"></div><br>
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<p><br></p>
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<p><br></p>
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<div class="related">
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<a href="../pythagoras.html">Pythagorean Theorem</a>
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<a href="pythagorean-theorem-proof.html">Pythagorean Theorem Algebraic Proof</a>
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<a href="pythagoras-3d.html">Pythagorean Theorem in 3D</a>
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<a href="index.html">Geometry Index</a>
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