Question

The circumference of a circle circumscribing an equilateral triangle is $24\pi$ units. The area of the circle inscribed in the equilateral triangle is $4a\pi$.Find $a$

Answer 1

$2\pi R=24\pi$(R is the radius of circumcircle)
$R=12$
$\sin {30}^{\circ }=\frac{r}{R}$(r is the radius of incircle)
$r=\frac{12}{2}=6$
$\therefore$ Area of incircle$=\pi r^2=36\pi$ comparing it with $4a\pi$ we get $4a=36$
Therefore, $a=9$
Ans: 9