The circumference of a circle circumscribing an equilateral triangle is `24piu n i t sdot` Find the area of the circle inscribed in the equilateral triangle.

Answer

`36pi` sq. unit

Explanation

In the figure, circumcircle of equilateral triangle ABC has circumference `24pi`.

Let R be the radius of the circle and O be the centre.
`:. 2piR=24pi`
`rArr R=12`
Circle inscribed in triangle touches BC at D, which is midpoint of BC.
In triangle ODB, `sin30^(@)=(OD)/(OB)=r/R` (r is the radius of incircle)
`:. r=12/2=6`
`:." Area of inscrived circle "=pir^2=36pi" sq. unit"`