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<!doctype html>
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<title>Pythagorean Triples</title>
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<h1 class="center">Pythagorean Triples</h1>
<p>A "Pythagorean Triple" is a set of positive <a href="whole-numbers.html">integers</a>, <b>a</b>, <b>b</b> and <b>c</b> that fits the rule:</p>
<p class="largest" align="center">a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup></p>
<div class="example">
<h3>Example: The smallest Pythagorean Triple is 3, 4 and 5.</h3>
<p>Let's check it:</p>
<p class="center larger">3<sup>2</sup> + 4<sup>2</sup> = 5<sup>2</sup></p>
<p>Calculating this becomes:</p>
<p class="center larger">9 + 16 = 25</p>
<p>Yes, it is a Pythagorean Triple!</p>
</div>
<h2>Triangles</h2>
<p style="float:left; margin: 0 10px 5px 0;"><img src="geometry/images/pythagoras-abc.svg" alt="Pythagoras"></p>
<p>When a triangle's sides are a Pythagorean Triple it is a <a href="right_angle_triangle.html">right angled triangle</a>.</p>
<p class="center"><img src="geometry/images/pythagoras-squares.svg" alt="pythagoras squares a^2 + b^2 = c^2"></p>
<p>See <a href="pythagoras.html">Pythagoras' Theorem</a> for more details.</p>
<div style="clear:both"></div>
<div class="example">
<h3>Example: The Pythagorean Triple of 3, 4 and 5 makes a Right Angled Triangle:</h3>
<p class="center"><img src="geometry/images/triangle-3-4-5-pyth.svg" alt="triangle 3 4 5 "></p>
</div>
<p class="center"><img src="geometry/images/triangle-3-4-5-leg.jpg" alt="triangle 3 4 5 lego" height="216" width="200"></p>
<p>&nbsp;</p>
<p>Here are two more Pythagorean Triples:</p>
<div class="clear">
<table align="center">
<tbody>
<tr class="large" align="center">
<td>5, 12, 13</td>
<td style="width:30px;">&nbsp;</td>
<td>9, 40, 41</td>
</tr>
<tr style="text-align:center;">
<td>&nbsp;</td>
<td>&nbsp;</td>
<td>&nbsp;</td>
</tr>
<tr style="text-align:center;">
<td><img src="geometry/images/triangle-5-12-13.svg" alt="5, 12, 13 Triangle"></td>
<td>&nbsp;</td>
<td><img src="geometry/images/triangle-9-40-41.svg" alt="9, 40, 41 Triangle"></td>
</tr>
<tr style="text-align:center;">
<td>5<sup>2</sup> + 12<sup>2</sup> = 13<sup>2</sup></td>
<td>&nbsp;</td>
<td>9<sup>2</sup> + 40<sup>2</sup> = 41<sup>2</sup></td>
</tr>
<tr style="text-align:center;">
<td>25 + 144 = 169</td>
<td>&nbsp;</td>
<td>(try it yourself)</td>
</tr>
</tbody></table>
</div>
<p>&nbsp;</p>
<p class="center larger">And each triangle has a right angle!</p>
<h2>List of the First Few</h2>
<p>Here is a list of the first few Pythagorean Triples (<b>not</b> including "scaled up" versions mentioned below):</p>
<div class="simple">
<table width="100%" border="0">
<tbody>
<tr style="text-align:center;">
<td>(3, 4, 5)</td>
<td>(5, 12, 13)</td>
<td>(7, 24, 25)</td>
<td>(8, 15, 17)</td>
</tr>
<tr style="text-align:center;">
<td>(9, 40, 41)</td>
<td>(11, 60, 61)</td>
<td>(12, 35, 37)</td>
<td>(13, 84, 85)</td>
</tr>
<tr style="text-align:center;">
<td>(15,112,113)</td>
<td>(16, 63, 65)</td>
<td>(17,144,145)</td>
<td>(19,180,181)</td>
</tr>
<tr style="text-align:center;">
<td>(20, 21, 29)</td>
<td>(20, 99,101)</td>
<td>(21,220,221)</td>
<td>(23,264,265)</td>
</tr>
<tr style="text-align:center;">
<td>(24,143,145)</td>
<td>(25,312,313)</td>
<td>(27,364,365)</td>
<td>(28, 45, 53)</td>
</tr>
<tr style="text-align:center;">
<td>(28,195,197)</td>
<td>(29,420,421)</td>
<td>(31,480,481)</td>
<td>(32,255,257)</td>
</tr>
<tr style="text-align:center;">
<td>(33, 56, 65)</td>
<td>(33,544,545)</td>
<td>(35,612,613)</td>
<td>(36, 77, 85)</td>
</tr>
<tr style="text-align:center;">
<td>(36,323,325)</td>
<td>(37,684,685)</td>
<td colspan="2">... infinitely many more ...</td>
</tr>
</tbody></table>
</div>
<h2>Scale Them Up</h2>
<p>The simplest way to create further Pythagorean Triples is to scale up a set of triples.</p>
<div class="example">
<h3>Example: scale <b>3, 4, 5</b> by 2 gives <b>6, 8, 10</b></h3>
<p>Which also fits the formula a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup>:</p>
<p class="center larger">6<sup>2</sup> + 8<sup>2</sup> = 10<sup>2</sup></p>
<p class="center larger">36 + 64 = 100</p>
</div>
<p>If you want to know more about them read <a href="numbers/pythagorean-triples.html">Pythagorean Triples - Advanced</a></p>
<p>&nbsp;</p>
<div class="questions">
<script>getQ(619,3252,777,3253,778,620,2242,2243,2244,3254);</script>&nbsp;
</div>
<div class="related">
<a href="pythagoras.html">Pythagoras Theorem</a>
<a href="right_angle_triangle.html">Right Angled triangles</a>
<a href="numbers/pythagorean-triples.html">Pythagorean Triples - Advanced</a>
<a href="triangle.html">Triangles</a>
<a href="proof180deg.html">Proof that a Triangle has 180°</a>
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