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I am looking for sliver polygons and am using the following formula to identify which polygons have a smaller area-to-circumference ratio (aka Thinness Ratio):

4 * pi * area/(length*length)

That much I understand. But what is not fully clear, is the 4 * Pi bit and why the length has to be squared. Can someone explain this in simple terms?

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    I believe that should be perimeter * perimeter rather than length * length. For a circle, the value is 1. When you think about it in terms of area/squared perimeter it starts to make sense. Compare a square that is 5x5 to a rectangle that is 9x1, both having a perimeter of 20, but the square having an area nearly 3 times bigger than the thinner rectangle. You can derive a similar result with calculus. – John Powell Jun 23 '15 at 14:04
  • Length is the circumference of the Polygon - so the same as perimeter – Robert Buckley Jun 23 '15 at 14:05
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    Perimeter would be the preferred terminology in mathematics, I would think. The point, anyway, is that an area reaches its maximum when a shape is regular, and falls rapidly as the sides become less equal in length, assuming a constant perimeter. – John Powell Jun 23 '15 at 14:06
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    You ask the same question in StackExchange Mathematics and the comment give you an explanation. This ratio is also known as Circularity ratio – gene Jun 23 '15 at 15:54
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    Cross-posting on SE sites is discouraged. Please review the SE meta document on the subject: meta.stackexchange.com/q/64068. The best way to resolve the issue would be to choose which site you would like to keep your question and delete the other one. – Aaron Jun 23 '15 at 16:08
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The idea behind the thinness ratio is that once known the perimeter of the unknown shape if the shape is similar to a circle the measured area should be equal to the theoretical area of a circle with the circumference equal to the perimeter of the unknown shape.

Knowing that the area of the circle is pi*r**2 and that the perimeter p is 2*pi*r hence r = p /2*pi substituting r in the formula of the area we obtain that the theoretical area of the circle with perimeter p is (p*p)/4*pi. Hence the ratio of the measured area vs theoretical is: (4*pi*Ames)/(p*p).

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