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198 lines
8.2 KiB
HTML
198 lines
8.2 KiB
HTML
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<title>Area of Irregular Polygons</title>
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<div id="content" role="main"><!-- #BeginEditable "Body" -->
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<h1 align="center">Area of Irregular Polygons</h1>
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<h2>Introduction</h2>
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<p>I just thought I would share with you a clever technique I once used to find the area of general <a href="polygons.html"> polygons</a>.</p>
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<p>The polygon could be <b>regular</b> (all angles are equal and all sides are equal) or <b> irregular</b></p>
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<table width="60%" border="0" align="center">
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<tr align="center">
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<td><b><img src="images/pentagon-regular.svg" alt="regular" /></b></td>
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<td><b><img src="images/pentagon-irregular.svg" alt="irregular" /></b></td>
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</tr>
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<tr align="center">
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<td>Regular</td>
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<td>Irregular</td>
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</tr>
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</table>
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<h3>The Example Polygon</h3>
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<p>Let's use this polygon as an example:</p>
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<p align="center"><img src="images/area-irregular-polygon1.gif" width="181" height="159" alt="irregular polygon" /></p>
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<h3>Coordinates</h3>
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<p>The first step is to turn each vertex (corner) into a <a href="../data/cartesian-coordinates.html">coordinate</a>, like on a graph:</p>
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<p align="center"><img src="images/area-irregular-polygon2.gif" width="295" height="258" alt="irregular polygon coords" /></p>
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<h3>Area Under One Line Segment</h3>
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<p>Now, for each line segment, work out the area <b>down to the x-axis</b>.</p>
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<p align="center"><img src="images/area-irregular-polygon3.gif" width="299" height="254" alt="irregular polygon coords and area of one graph trapezoid" /></p>
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<p><b>So, how do we calculate each area?</b></p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/area-trapezoid.svg" alt="area of graph trapezoid is width by average height" /> </p>
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<h3>Average the two heights, then multiply by the width</h3>
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<p>Example: For the shape highlighted above, we take the two heights (the "y" coordinates 2.28 and 4.71) and work out the average height:</p>
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<p align="center" class="larger"> (2.28+4.71)/2 = 3.495</p>
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<p>Work out the width (the difference between the "x" coordinates 2.66 and 0.72)</p>
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<p align="center" class="larger">2.66-0.72 = 1.94</p>
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<p>The area is width×height:</p>
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<p align="center" class="larger">1.94 × 3.495 = 6.7803</p>
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<p> </p>
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<h3>Add Them All Up</h3>
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<p>Now add them all up! </p>
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<p>But the trick is to add when they go forwards (positive width), and subtract when they go backwards (negative width). </p>
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<p>If you always go clockwise around the polygon, and always subtract the first "x" coordinate from the second, it works out naturally, like this:</p>
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<div class="simple">
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<table border="0" align="center">
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<tr align="right">
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<th colspan="2" align="center">From</th>
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<th colspan="2" align="center">To</th>
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<td bgcolor="#FFFFFF"> </td>
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<td bgcolor="#FFFFFF"> </td>
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<td bgcolor="#FFFFFF"> </td>
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</tr>
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<tr align="right">
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<th width="50">x</th>
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<th width="50">y</th>
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<th width="70">x</th>
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<th width="50">y</th>
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<th width="90">Avg Height</th>
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<th width="90">Width (+/-)</th>
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<th width="90">Area (+/-)</th>
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</tr>
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<tr align="right">
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<td>0.72</td>
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<td>2.28</td>
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<td>2.66</td>
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<td>4.71</td>
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<td>3.495</td>
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<td>1.94</td>
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<td>6.7803</td>
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</tr>
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<tr align="right">
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<td>2.66</td>
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<td>4.71</td>
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<td>5</td>
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<td>3.5</td>
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<td>4.105</td>
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<td>2.34</td>
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<td>9.6057</td>
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</tr>
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<tr align="right">
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<td>5</td>
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<td>3.5</td>
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<td>3.63</td>
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<td>2.52</td>
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<td>3.01</td>
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<td>-1.37</td>
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<td>-4.1237</td>
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</tr>
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<tr align="right">
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<td>3.63</td>
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<td>2.52</td>
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<td>4</td>
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<td>1.6</td>
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<td>2.06</td>
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<td>0.37</td>
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<td>0.7622</td>
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</tr>
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<tr align="right">
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<td>4</td>
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<td>1.6</td>
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<td>1.9</td>
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<td>1</td>
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<td>1.3</td>
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<td>-2.1</td>
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<td>-2.7300</td>
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</tr>
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<tr align="right">
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<td>1.9</td>
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<td>1</td>
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<td>0.72</td>
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<td>2.28</td>
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<td>1.64</td>
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<td>-1.18</td>
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<td>-1.9352</td>
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</tr>
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<tr align="right">
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<td bgcolor="#FFFFFF"> </td>
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<td bgcolor="#FFFFFF"> </td>
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<td bgcolor="#FFFFFF"> </td>
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<td bgcolor="#FFFFFF"> </td>
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<td bgcolor="#FFFFFF"> </td>
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<td bgcolor="#FFFFFF">Total:</td>
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<th><b>8.3593</b></th>
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</tr>
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</table>
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<p>You can also go the other direction. If you get a negative area just make it positive.</p>
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<p>And it looks like this:</p>
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<p align="center"><img src="images/area-irregular-polygon4.gif" width="303" height="256" alt="irregular polygon area all trapezoids" /></p>
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<p>So that's it! The area is<b> 8.3593</b></p>
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<h2>Area of Polygon Tool</h2>
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<p>Glad you read this far! You are rewarded with a link to the <a href="area-polygon-drawing.html">Area of a Polygon Drawing Tool</a> that can do all of this for you. It also accepts manual entry of coordinates.</p>
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getQ(5288, 5289, 5290, 5291, 5292, 5293, 5294, 5295, 1762, 1763);
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</script> </div>
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<div class="related"><a href="index.html">Geometry Index</a></div>
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