Question

A circle is inscribed in an equilateral triangle of side a. What is the area of any square inscribed in the circle?

(a) \(\frac{a^2}{3}\)

(b) \(\frac{a^2}{4}\)

(c) \(\frac{a^2}{6}\)

(d) \(\frac{a^2}{8}\)

Answer 1

(c) \(\frac{a^2}{6}\)

AB = BC = CA = a

∴ Height of the equilateral Δ = \(\frac{\sqrt3}{2}\times a\)

O being the centroid of the Δ ABC,

3OD = \(\frac{\sqrt3}{2}a\) ⇒ OD = \(\frac{\sqrt3}{6}a\)

Diagonal of the square = 2 × OD

= 2 x \(\frac{\sqrt3}{6}a\) = \(\frac{\sqrt3}{2}a\)

= \(\frac{1}{\sqrt3}a\)

∴ Area of the square = \(\frac12\) (diagonal)²

= \(\frac12\) x (\(\frac{1}{\sqrt3}a\))² = \(\frac16\)a².