2003 AMC 10A Problem 17

Question

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?

\dfrac{3\sqrt{2}}{\pi}

\dfrac{3\sqrt{3}}{\pi}

\sqrt{3}

\dfrac{6}{\pi}

\sqrt{3}\pi

Answer: B

Concepts:equilateral triangle circumcircle, circumcenter, and circumradius special right triangle

Difficulty rating: 1600

Small hint:

For an equilateral triangle of side

s,

the circumradius is

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

Big hint:

Set the perimeter

3s

equal to the area

\pi R^2,

then substitute

s = R\sqrt{3}

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}},

so

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,

so

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

Thus, the correct answer is B.