# The radius of a circle inscribed in an equilateral triangle is 2. What is the perimeter of the triangle?

Perimeter equals to

There are many ways to address this problem. Here is one of them.

The center of a circle inscribed in to a triangle lies on intersection of its angles' bisectors. For equilateral triangle this is the same point where its altitudes and medians intersect as well.

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Perimeter equals to

Alternative method is below.

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The radius ( r ) of a circle inscribed in an equilateral triangle is related to the side length ( s ) of the equilateral triangle by the formula ( r = \frac{s\sqrt{3}}{6} ).

Given that ( r = 2 ), we can solve for the side length ( s ): [ 2 = \frac{s\sqrt{3}}{6} ] [ 12 = s\sqrt{3} ] [ s = \frac{12}{\sqrt{3}} = 4\sqrt{3} ]

The perimeter ( P ) of the equilateral triangle is three times the side length: [ P = 3s = 3(4\sqrt{3}) = 12\sqrt{3} ]

So, the perimeter of the equilateral triangle is ( 12\sqrt{3} ).

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