Circle inscribed in a rhombus

Text

A rhombus has area $50\sqrt3$ cm$^2$ and an interior angle of $60^\circ$. Find the area of the circle inscribed in the rhombus.

Solution

Let $\ell$ be the side; the area of the rhombus is $\ell^2\sin60^\circ$: $\ell^2\frac{\sqrt3}{2}=50\sqrt3\ \Rightarrow\ \ell^2=100\ \Rightarrow\ \ell=10.$ The radius of the inscribed circle is the apothem of the rhombus, i.e. area over semiperimeter: $r=\frac{50\sqrt3}{2\cdot10}=\frac{5\sqrt3}{2}.$ Area of the circle: $\pi r^2=\pi\left(\frac{5\sqrt3}{2}\right)^2=\frac{75}{4}\pi\approx58.90\ \text{cm}^2.$ $\boxed{\mathcal A=\tfrac{75}{4}\pi\approx58.90\ \text{cm}^2}$