Q: 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?

A: 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). 

A: The Thinness Ratio describes the relation between a polygons perimeter to its area using geometric attributes of a circle as a basis for comparison. A definition can be found in  Microscope Image Processing, page 201. For a circle, we know that
A = pi * r^2

and
p = 2 * pi * r

Solving the perimeter equation for r gives:
r = p / (2 * pi)

which can then be substituted into the area equation to give:
A = pi * (p / (2 * pi))^2 = pi * p^2 / (4 * pi^2)
  = p^2 / (4 * pi)

Collecting all terms on one side, we then get:
A * 4 * pi / (p^2) = 1

which is a true statement for circles. We then define the Thinness Ratio (TR) to be:
TR(A,p) = A * 4 * pi / (p^2)

When the polygon in question is a circle, we know that TR = 1. Otherwise, TR gives us a measure of how "squashed" or non-circular the polygon in question is. In fact, the book I referenced earlier defines a polygon's Circularity as the reciprocal of the Thinness Ratio and these concepts are linked. In practice, we use the Thinness Ratio because we can readily compute a polygon's perimeter and area.
