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