Area inside Astroid

Theorem
The area inside an astroid constructed within a circle of radius $a$ is given by:
 * $\mathcal A = $

Proof

 * AstroidArea.png

By symmetry, it is sufficient to evaluate the area shaded yellow and to multiply it by $4$.

By Equation of Astroid:

Thus:
 * $\displaystyle \mathcal A = \int_0^a \, \mathrm d x$