Area inside Astroid

Theorem
The area inside an astroid $H$ constructed within a circle of radius $a$ is given by:
 * $\mathcal A = \dfrac {3 \pi a^2} 8$

Proof
Let $H$ be embedded in a cartesian coordinate plane with its center at the origin and its cusps positioned on the axes.


 * AstroidArea.png

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

By Equation of Astroid:


 * $\begin{cases}

x & = a \cos^3 \theta \\ y & = a \sin^3 \theta \end{cases}$

Thus:

Simplifying the integrand:

Thus: