<s>[INST] Area of a circle inscribed in a rhombus? Let's say we have a rhombus with diagonals $a$ and $b$, which contains an inscribed circle. How can we find the area of that circle in terms of $a$ and $b$?
 [/INST] 
Obviously, the radius of inscribed circle is also a height $h=OH$ of the right triangle $\triangle AOB$. To find it, use equations for triangle's area
$$
S_{\triangle AOB} = \frac 12 \frac a2 \frac b2 = \frac {ab}8 = \frac 12 ch
$$
where $c = AB$ is a hypotenuse. So $r = h = \frac {ab}{4c} = \frac {ab}{4\sqrt{\frac {a^2}4+\frac {b^2}4}} = \frac {ab}{2\sqrt{a^2+b^2}}$, and therefore
$$
S_c = \pi r^2 = \frac {\pi a^2 b^2}{4\left (a^2+b^2 \right )}
$$
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