The area of a circle whose circumference is 24π is kπ. What is the value of k?

2010 AMC 10A Problem 5

Attempt Problem 5 of the 2010 AMC 10A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2010 AMC 10A solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

The area of a circle whose circumference is

24\pi

k\pi.

What is the value of

k?

6

12

24

36

144

Use the circumference to find the radius first

From

2\pi r=24\pi

, the desired

k

r^2

Recall that the formula for the circumference of a circle is

2 \pi r.

We then have that

24\pi = 2\pi r

r = 12.

The area of a circle is

\pi r^2,

so we have that

k \pi = \pi 12^2

k = 144.

Thus, E is the correct answer.