2010 AMC 10A Problem 5

P1-10~1 minPrint

Circles Set up the equation/formula and compute; no special trick needed

The area of a circle whose circumference is $24\pi$ is $k\pi$ . What is the value of $k$ ?

Show solution without answering

Key insight

Go through the radius: the circumference 2 times pi times r equals 24 pi, so r = 12, and the area is pi times 12 squared = 144 pi.

Solution

Both the circumference and the area are built from the radius, so find the radius first.

The circumference of a circle with radius $r$ is $2\pi r$ .
We are told it is $24\pi$ , so $2\pi r = 24\pi$ . Divide both sides by $2\pi$ : $r = 12$ .
The area is $\pi r^2 = \pi \times 12 \times 12 = 144\pi$ .
The area is written as $k\pi$ , so $k = 144$ .

The answer is $\boxed{\textbf{(E)}\ 144}$ .

Why this works

For circles, the radius is the key that connects everything: diameter, circumference and area. Turn whatever you are given into the radius, then turn the radius into what is asked. Leaving $\pi$ as a symbol saves you from any decimals.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

The circumference of a circle of radius $r$ is $2\pi r$ . Setting $2\pi r = 24\pi$ gives $r = 12$ .

The area is then
$\pi r^2 = \pi \cdot 12^2 = 144\pi,$
so $k = 144$ .

The answer is $\boxed{\textbf{(E)}\ 144}$ .

Review in the AoPS books

Introduction to Geometry, Chapter 11: Circles
Introduction to Geometry, Chapter 12: Circles and Angles
Introduction to Geometry, Chapter 13: Power of a Point