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2010 AMC 10A Problem 5

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2010 AMC 10A Problem 5AMC 10 Step by Step

Name: Date: September 13, 2026

Circles · difficulty 1 of 5 · about 1 min · position P1-10

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

A)
$6$
B)
$12$
C)
$24$
D)
$36$
E)
$144$

Answer

Key insight

Circumference 2 pi r = 24 pi gives r = 12, so the area is pi r^2 = 144 pi.

Solution

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 $\boxed{\textbf{(E)}\ 144}$ .

Why this works

Every circle question routes through the radius: convert the given quantity (here the circumference) to $r$ , then convert $r$ to what is asked. Keeping the $\pi$ symbolic avoids decimal work entirely.

The trap

Using 24 as the radius (or the diameter as the radius) and reporting 24 or 36 as k.

Common mistakes

Using 24 as the radius (or the diameter as the radius) and reporting 24 or 36 as k.
Halving instead of squaring at the last step and answering $12$ .

Techniques

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