Inscribed Circle of Equilateral Triangle

The circumference of a circle circumscribing an equilateral triangle is 24\pi units. Find the number of square units in the area of the circle inscribed in the same triangle.
Source: NCTM Mathematics Teacher, February 2006

SOLUTION
image
In equilateral triangle ABC, angle bisector BM is also the perpendicular bisector of AC. That makes point I the circumcenter, the incenter, and the centroid of triangle ABC.
24\pi=2\pi\times IB
IB=12
IM=IB/2=6 because I is the centroid
Area of incircle
\pi\times IM^2=36\pi

Answer: 36\pi square units

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About mvtrinh

Retired high school math teacher.
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