To what extend a polygon can be considered a circle?

Question

I have a polygon of which I know:

• Area
• $x_{\max}$, $x_{\min}$
• $y_{\max}$, $y_{\min}$

and I would like to establish to what extend the polygon can be considered a circle.

From what I found, for calculating the circularity or the compactness I need to know the perimeter, which I don't have. So far, my idea is to calculate the roundness:

    4 * Area / ( Pi * Major axis)^2

Is there anything else I could calculate?

Answer 1

How about comparing $x_{\max}-x_{\min}$ with $y_{\max}-y_{\min}$?

We can say the followings :

• If it is a circle, then $x_{\max}-x_{\min}=y_{\max}-y_{\min}$.
• If $x_{\max}-x_{\min}\not=y_{\max}-y_{\min}$, then it is not a circle.

Also, how about comparing $\pi$ with $\frac{\text{area}}{\left(\frac{x_{\max}-x_{\min}}{2}\right)^2}$?

We can say the followings :

• If it is a circle, then $\pi=\frac{\text{area}}{\left(\frac{x_{\max}-x_{\min}}{2}\right)^2}$.

• If $\pi\not =\frac{\text{area}}{\left(\frac{x_{\max}-x_{\min}}{2}\right)^2}$, then it is not a circle.