Question

If a circle is inscribed in an equilateral triangle of side a, then area of the square inscribed in the circle is

• A. $\frac{a^2}{6}$
• B. $\frac{a^2}{3}$
• C. $\frac{2a^2}{5}$
• D. $\frac{2a^2}{3}$

Answer 1

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

The radius of the circle in terms of the side of the triangle is
$\tan {30}^0.\frac{a}{2}$
$R=\tan {30}^0.\frac{a}{2}$
$R=\frac{a}{\sqrt{3}.2}$
Now let the side of the square be $a$.
Now the diameter of the circle will be the diagonal of the square.
Applying Pythagoras theorem we get
$\sqrt{2a^{\prime 2}}=2R$
$a^{\prime 2}=2R^2$
$a^{\prime 2}=\frac{2.a^2}{4.3}$
$a^{\prime 2}=\frac{a^2}{6}$.
Hence area is $=\frac{a^2}{6}$.