The area of a circle inscribed in an equilateral triangle is 154 cm². Find the perimeter of the triangle :

Solution(By Examveda Team)

Radius of incircle $$ = \frac{a}{{2\sqrt 3 }}$$
Area of incircle $$ = \left( {\frac{{\pi \times {a^2}}}{{12}}} \right)c{m^2}$$
$$\eqalign{ & \therefore \frac{{\pi {a^2}}}{{12}} = 154 \cr & \Rightarrow {a^2} = \frac{{154 \times 12 \times 7}}{{22}} \cr & \Rightarrow a = 14\sqrt 3 \cr} $$
∴ Perimeter of the triangle :
$$\eqalign{ & = \left( {3 \times 14\sqrt 3 } \right)cm \cr & = \left( {42 \times 1.732} \right)cm \cr & = 72.7\,cm\,(\text{approx}) \cr} $$