2003 AMC 10A Problem 17

Problem 17 of 25IntermediateGeometry

The number of inches in the perimeter of an equilateral triangle equals the number of square inches in the area of its circumscribed circle. What is the radius, in inches, of the circle?

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Solution

Let the side length be

s

and the circumradius be

R.

From a

30

60

90

triangle formed by the center and a side,

R = \dfrac{s}{\sqrt{3}},

s = R\sqrt{3}.

The perimeter is

3s = 3R\sqrt{3}

and the circle’s area is

\pi R^2.

Setting them equal,

3R\sqrt{3} = \pi R^2,

R = \dfrac{3\sqrt{3}}{\pi}.

Thus, the correct answer is B.

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