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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) \(\frac{a^2}{8}\)

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(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 (\(\frac{1}{\sqrt3}a\))\(\frac16\)a2.

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