Question

A circle is inscribed in an equilateral triangle of side $a$. The area of any square inscribed in this circle is... .

Answer 1

In an equilateral triangle, the radius of incircle $=\frac{1}{3}\times$ median of the triangle
$=\frac{1}{3}\sqrt{a^2-\frac{a^2}{4}}=\frac{1}{3}\sqrt{\frac{4a^2-a^2}{4}}=\frac{a}{2\sqrt{3}}$
Therefore, area of the square inscribed in this circle $=2(\text{ radius of circle }{)}^2=\frac{2a^2}{4\cdot 3}=\frac{a^2}{6}$ sq unit .