AMC 8 · 2012 · #24

Grade 7 geometry-2d

Archetype Compound Area by Decomposition / Difference └ Sub-archetype Circle-Polygon Hybrid

area-circlesarea-rectanglescoordinate-geometryspatial-visualization area-differenceidentify-subproblems ↑ Prerequisites: area-circlespythagorean-theoremcoordinate-geometry

📏 Long solution 💡 4 insights 📊 Diagram

Problem

A circle of radius 2 is cut into four congruent arcs (each a quarter-circle). The four arcs are rejoined into a four-pointed "star" whose sides curve inward. Find the ratio of the area of this star to the area of the original circle.

Pick an answer.

(A)

$\frac{4-\pi}{\pi}$

(B)

$\frac{1}{\pi}$

(C)

$\frac{\sqrt{2}}{\pi}$

(D)

$\frac{\pi-1}{\pi}$

(E)

$\frac{3}{\pi}$

AMC 8 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

The star is an odd, curved shape — we cannot apply any single area formula to it directly. Tool #1 (Draw a Diagram) is the right first move: by sketching the star inside a square of side 4 whose corners coincide with the star's points, we can SEE that the four "bites" taken out of the square are exactly four quarter-circles of radius 2 — which together form one full circle of radius 2. Tool #7 (Identify Subproblems) then turns the problem into two easy pieces we already know: (a) area of a 4 × 4 square, and (b) area of a circle of radius 2. The star is (a) minus (b), and the ratio falls out by simple division.

1STEP 1

Place the star's four points on the axes and box it in the square x,y = ±2: side 4, so its area is 16.

Bounding square side = 4, A_square = 4 × 4 = 16

2STEP 2

Each of the four square corners (two sides plus one inward arc) is exactly a quarter-disk of radius 2, area π.

Each corner piece =

\frac{1}{4}

π (2)² = π

3STEP 3

The four quarter-disks reassemble into one full circle of radius 2 — the original circle — total area 4π.

4 ×

\frac{1}{4}

π(2)² = π(2)² = 4π

4STEP 4

Star area = square minus the four corners = 16 - 4π.

A_star = 16 - 4π

5STEP 5

Divide by the circle's area 4π and factor 4 from the top: $\frac{16 - 4π}{4π}$ = $\frac{4-π}{π}$ , choice (A).

\frac{A_star}{A_circle}

\frac{16 - 4π}{4π}

\frac{4(4 - π)}{4π}

\frac{4 - π}{π}

→ (A)

Answer

\frac{4-π}{π}

Numerically, π ≈ 3.14, so

\frac{4 - π}{π}

≈

\frac{0.86}{3.14}

≈ 0.27. That says the star takes up about 27% of the original circle's area — which matches the picture: the star is clearly much smaller than the original circle (more than half of the circle's area sits outside the star in those four "crescent" gaps). Also, all five answer choices have π in the denominator, which is consistent with our setup (we divided by 4π).

💡Key takeaway

Draw a square around the star: the four "bites" out of the square fit together to make the original circle, so the star is just square minus circle — Grade 7 circle-area reasoning does the rest.

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