Question

A circle is inscribed in an equilateral triangle and a square is inscribed in the circle. The ratio of the area of the triangle to that of the square is

• A. $\sqrt{3}:1$
• B. $\sqrt{3}:\sqrt{2}$
• C. $3\sqrt{3}:2$
• D. $3\sqrt{3}:\sqrt{2}$

Answer 1

The correct option is C $3\sqrt{3}:2$

Area of triangle $=\frac{\sqrt{3}}{4}{a}^2$ (where a is one side of the triangle)
Radius R of the circle $=\frac{a}{2}\tan {30}^{\circ }$
$=\frac{a}{2}\times \frac{1}{\sqrt{3}}=\frac{a}{2\sqrt{3}}$
$\therefore$ Diameter of the circle $=\frac{a}{\sqrt{3}}$
$\therefore$ Diagonal of the square $=\frac{a}{\sqrt{3}}$
$\therefore$ Area of the inscribed square $=\frac{1}{2}\times {\left(\frac{a}{\sqrt{3}}\right)}^2=\frac{a^2}{6}$
$\therefore$ Ratio of the area of the triangle to that of the square
$=\frac{\sqrt{3}{a}^2}{4}:\frac{a^2}{6}=3\sqrt{3}:2$