# PrA

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PrA is a simple ad-hoc parameter to define the "non-circularity" or eccentricity of a 2D object. This quantity is simply:

{\displaystyle {\begin{alignedat}{2}\mathrm {PRA} ={\frac {Pr}{A}}\end{alignedat}}}

Where ${\displaystyle P}$ is the object's perimeter, ${\displaystyle A}$ is its surface area, and ${\displaystyle r}$ is an effective size (radius), computed based on the corresponding circle of the same area:

{\displaystyle {\begin{alignedat}{2}r={\sqrt {\frac {A}{\pi }}}\end{alignedat}}}

This definition of PrA is convenient, since it provides a simple measure of eccentricity. In particular, for a circle one expects:

{\displaystyle {\begin{alignedat}{2}\mathrm {PRA} &={\frac {Pr}{A}}\\&={\frac {(2\pi r)(r)}{\pi r^{2}}}\\&=2\end{alignedat}}}

Since a circle has the minimal perimeter (for a given area), this is a limiting value of PrA:

{\displaystyle {\begin{alignedat}{2}\mathrm {PRA} \geq 2\end{alignedat}}}

And thus any non-circular object will have a larger PrA.