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The circumference of a circle circumscri...

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

Text Solution

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The correct Answer is:
`36pi` sq. unit
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"`
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