Question

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

Answer 1

Circumference of circle $=24\pi$
$2\pi R=24\pi$
$R=12$
$\therefore OB=OC=AO=12$
Centre of circle is centroid of $△ABC$ [equilateral]
Centroid divides AD in the ratio of 2:1
$AO=\frac{2}{3}AD$
$12=\frac{2}{3}AD$
$AD=18$
$\therefore OD=6$
Radius of circle inscribed in equilateral $△ABC=OD$
i.e. $r=6$
$\therefore Area=\pi r^2$
$=36\pi$ sq.units.