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

Seven unit circles are packed inside one large circle: one circle sits at the center, six more ring it and touch both the center circle and the large outer circle. Find the area of the shaded part of the large circle that none of the seven small circles covers.

Pick an answer.

(A)

$\pi$

(B)

$1.5\pi$

(C)

$2\pi$

(D)

$3\pi$

(E)

$3.5\pi$

How to solve

Strategy Change Focus / Count the Complement

The shaded region has a jagged, six-lobed boundary that would be miserable to measure directly. Tool #16 (Count the Complement) sidesteps that entirely: the shaded area is just the whole large disk minus the seven round holes punched out of it, and a disk's area is easy. Before subtracting, tool #1 (Draw a Diagram) reads the tangency chain off the picture to pin down the one missing number — the large radius. Tool #7 (Identify Subproblems) then splits the job into two clean pieces: the big circle's area and the total area of the seven identical small circles. The main trap is miscounting the small circles as six (the visible ring) and forgetting the center one, which gives the decoy (D) 3π.

1STEP 1

Build the large radius

Centre to ring centre is 1+1=2, and one more unit reaches the rim, so the large radius is 3.

R = (1+1)_center to ring center + 1_ring radius = 3

2STEP 2

Area of the large circle

The large disk covers π × 3² = 9π, the budget the holes come out of.

π R² = π(3)² = 9π

3STEP 3

Total area of the seven holes

One centre plus six ring circles is seven, each of area π, so together 7π.

7 × π(1)² = 7π

4STEP 4

Subtract the holes

Subtract the holes: 9π - 7π = 2π, choice (C).

9π - 7π = (9-7)π = 2π → (C)

Answer

2π

The seven small circles cover 7π out of the large circle's 9π, so only 2π — a bit over a fifth of the disk — stays shaded. That fits the picture, where the circles crowd nearly the whole interior and leave just thin curved slivers between them. The answer 2π is smaller than every circle-total in play, which is what an 'uncovered leftover' should be, and it is positive, as any real area must be.

💡Key takeaway

When a shape is a big region with holes punched out, find its area by taking the whole area and subtracting the holes — just be sure to count every hole.

Build the large radius
Area of the large circle
Total area of the seven holes
Subtract the holes

Build the large radius

Area of the large circle

Total area of the seven holes

Subtract the holes

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