AMC 10 · 2015 · #9

Grade 7 geometry-2d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Nested Frame / Concentric Difference
area-circlesarea-difference identify-subproblems ↑ Prerequisites: area-circles
📏 Short solution 💡 2 insights 📊 Diagram
Problem
A shaded figure (the shark's fin falcata) sits in the first quadrant. Its outer boundary is the quarter of the circle of radius 3 centered at (0,0), plus the segment from (0,0) to (3,0); a curved bite is taken out by the part of the small circle of radius 3/2 centered at (0,3/2) that lies in the first quadrant. Find the area of the shaded figure.

Pick an answer.

(A)
$\dfrac{4\pi}{5}$
(B)
$\dfrac{9\pi}{8}$
(C)
$\dfrac{4\pi}{3}$
(D)
$\dfrac{7\pi}{5}$
(E)
$\dfrac{3\pi}{2}$

AMC 10 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Identify Subproblems

The fin is an odd shape, but it is built from two friendly pieces, so Tool #7 (Identify Subproblems) splits it into a large quarter-circle and a small half-circle that gets removed. Tool #1 (Draw a Diagram) is what lets you see that the small circle's diameter sits on the y-axis, so its first-quadrant part is exactly a semicircle. Tool #16 (Change Focus / Count the Complement) finishes the job: instead of measuring the strange curved region directly, take the big quarter and subtract the bitten-out half-circle.

1STEP 1

Area of the big quarter-circle

The big circle (radius 3) has area 9π, and the corner quarter is one of four equal wedges, so it is 9π/4.

1/4 π (3)² = 9π/4
2STEP 2

The small bite is a semicircle

The small circle's diameter lies on the y-axis, so its first-quadrant part is a semicircle of area 9π/8.

1/2 π(3/2)² = 1/2·9π/4 = 9π/8
3STEP 3

Subtract the bite from the quarter

Subtract the semicircle from the quarter: 9π/4 - 9π/8 = 9π/8, which is choice (B).

9π/4 - 9π/8 = 18π/8 - 9π/8 = 9π/8 → (B)
Answer
9π/8
The two areas turned out equal-looking but not equal: the quarter is 9π/4=18π/8 and the removed semicircle is 9π/8, exactly half of it. So the fin should be half the quarter-circle's area, 9π/8≈ 3.53. That is positive and smaller than the quarter (≈ 7.07), which fits a region carved out of the quarter — and it matches choice (B).
💡Key takeaway

Break a weird shape into a big slice minus a small bite, then subtract the two areas.

  • Area of the big quarter-circle
  • The small bite is a semicircle
  • Subtract the bite from the quarter

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