Explanation of the Thinness ratio formula?

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?

• 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
• 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
• 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
• 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