AMC 10 · 2010 · #5

Grade 7 geometry-2d

Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps

area-circlesperimeterformula-substitution work-backwards ↑ Prerequisites: area-circles

📏 Short solution 💡 1 insight

Problem

A circle has circumference 24π. Its area can be written as kπ. Find the number k.

Pick an answer.

(A)

$\ 6$

(B)

$\ 12$

(C)

$\ 24$

(D)

$\ 36$

(E)

$\ 144$

AMC 10 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

The circumference and the area of a circle are not directly linked, but both are built from the same radius r. So name that radius with a letter (Tool #4, Introduce a Variable). The given circumference is the finished output of the formula C = 2π r; run that formula in reverse to recover r (Tool #11, Work Backwards). That splits the task into two clean subproblems (Tool #7): first find r from the circumference, then feed r into the area formula. Solving in this order means no guessing among the five choices.

1STEP 1

Name the radius

Let r be the radius. The circumference formula C = 2π r must equal the given 24π.

2π r = 24π

2STEP 2

Work back to the radius

Divide both sides of 2π r = 24π by 2π: the π cancels and 24 ÷ 2 gives r = 12.

r = 24π/2π = 12

3STEP 3

Compute the area

Put r = 12 into A = π r²: 12² = 144, so A = 144π, and matching kπ gives k = 144.

A = π r² = π (12)² = 144π → k = 144

Answer

144

Check the radius against the circumference: 2π (12) = 24π, which matches the given circumference exactly. A radius of 12 is fairly large, so an area of 144π (much bigger than the circumference number 24) makes sense because area grows with the square of the radius. Choice (E) 144 is the only option equal to 12², which is a strong sign it is right.

💡Key takeaway

Both the circumference and the area come from the radius, so back the radius out of the circumference first, then square it in the area formula to get k = 144.

Name the radius
Work back to the radius
Compute the area

Name the radius

Work back to the radius

Compute the area

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