AMC 10 · 2002 · #5

Grade 7 geometry-2d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Nested Frame / Concentric Difference
area-circlestangent-circles area-difference ↑ Prerequisites: area-circles
📏 Medium solution 💡 2 insights 📊 Diagram
Problem
A circle of radius 3 and a circle of radius 2 sit side by side, touching each other, and both fit exactly inside a larger circle that touches each of them. Find the area inside the big circle but outside the two small circles.

Pick an answer.

(A)
$3\pi$
(B)
$4\pi$
(C)
$6\pi$
(D)
$9\pi$
(E)
$12\pi$

AMC 10 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

The whole problem turns on one picture fact: the flat line drawn through both small centers is actually a diameter of the big circle, so Tool #1 (Draw a Diagram) unlocks the size of the big circle by reading the lengths straight off that line. Once the big radius is known, Tool #7 (Identify Subproblems) splits the shaded area into three easy circle-area calculations that combine by subtraction — big circle minus the two small ones.

1STEP 1

Read the big diameter off the line

The line through both small centers crosses the big circle end to end, so the big diameter is 3+3+2+2=10.

diameter = 2(3)+2(2)=10
2STEP 2

Get the big radius

Halve it: the radius is half the diameter, so the big circle has radius 10÷2=5.

R = 10/2=5
3STEP 3

Area of the big circle

Use the circle-area formula π r² with r=5: the big circle has area π · 5² = 25π.

π R² = π(5)² = 25π
4STEP 4

Area of the two small circles

Same formula for each small circle: π(3)²=9π and π(2)²=4π, so together they cover 13π.

π(3)²+π(2)² = 9π+4π = 13π
5STEP 5

Subtract to get the shaded area

Take the covered part away: 25π-13π=(25-13)π=12π, which is choice (E).

25π-13π=(25-13)π=12π → (E)
Answer
12π
The big circle (25π) is much larger than the two small ones combined (13π), so a shaded region of 12π — a little less than half the big circle — looks right for two chunks bitten out of it. A quick sanity check on the smaller choices: 3π or 4π would mean the small circles cover almost the whole big circle, which the picture clearly contradicts. The key move that must be right is the diameter: mistakenly using R=5 as a diameter or forgetting to double the radii would throw the whole answer off.
💡Key takeaway

Line the two circles up edge to edge and their widths add to the big circle's diameter; then the shaded part is just the big circle's area minus the two smaller ones.

  • Read the big diameter off the line
  • Get the big radius
  • Area of the big circle
  • Area of the two small circles
  • Subtract to get the shaded area

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