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

The correct option is C $\frac{a^2}{6}$

In $△ABC$
We know sides of a equilateral are always equal
$\therefore AB=BC=CA=a$
Height of the equilateral,$\Delta =\frac{\sqrt{3}}{2}\times a$
O is the centre of the ABC
$3OD=\frac{\sqrt{3}}{2}a$
$\Rightarrow OD=\frac{\sqrt{3}}{6}$
Diagonal of the square$=2\times OD$
$=2\times \frac{\sqrt{3}}{6}a=\frac{\sqrt{3}}{3}a$
$=\frac{1}{\sqrt{3}}a$
$Areaofthesquare=\frac{1}{2}(diagonal{)}^2$
$=\frac{1}{2}\times {\left(\frac{1}{\sqrt{3}}a\right)}^2$
$=\frac{1}{6}{a}^2$