AMC 8 · 2014 · #20

Grade 7 geometry-2d

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

area-rectanglesarea-circlesestimation area-differenceidentify-subproblems ↑ Prerequisites: area-rectanglesarea-circles

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

A 3 × 5 rectangle ABCD has three circles centered at three of its corners: radius 1 at A, radius 2 at B, radius 3 at C. Only the parts of the circles that fall inside the rectangle matter. Find the area of the rectangle that is not covered by any circle, then pick the closest answer choice.

Pick an answer.

(A)

3.5

(B)

4.0

(C)

4.5

(D)

5.0

(E)

5.5

AMC 8 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The shaded region is the rectangle with three corner pieces removed, so the natural move is Tool #7 (Subproblems): compute the rectangle's area, compute each quarter-circle's area, then subtract. Tool #1 (Diagram) keeps us honest about which part of each circle sits inside the rectangle — at a corner with a 90° angle, exactly one quarter does — and lets us check on the picture that the quarter-circles do not overlap before we add them up.

1STEP 1

The rectangle's area is DA × CD = 5 × 3 = 15 — the whole we carve corners from.

Area_rect = 5 × 3 = 15

2STEP 2

Each 90° corner traps exactly a quarter-circle, so apply the area rule $\frac{1}{4}$ π r² to each circle.

Area_A =

\frac{1}{4}

π(1)² = 0.25π, Area_B =

\frac{1}{4}

π(2)² = π, Area_C =

\frac{1}{4}

π(3)² = 2.25π

3STEP 3

Overlap check: 1+2=3=AB and 2+3=5=BC, so the quarter-circles only touch and add with no double-count.

1 + 2 = 3 = AB, 2 + 3 = 5 = BC

4STEP 4

Add the three disjoint quarter-circles: 0.25π + π + 2.25π = 3.5π covered.

Total covered = 0.25π + π + 2.25π = 3.5π

5STEP 5

Uncovered = 15 - 3.5π ≈ 15 - 10.99 = 4.01, closest to choice (B) 4.0.

15 - 3.5π ≈ 15 - 3.5(3.14) = 15 - 10.99 = 4.01 → (B) 4.0

Answer

4.0

Sanity-check the magnitude. The rectangle has area 15. The three quarter-circles together are 3.5π ≈ 11, which is a bit more than two-thirds of the rectangle — visually that matches the figure, where the big circle at C alone eats up a large corner. So the leftover should be a small single-digit number, and 15 - 11 = 4 lands right in the middle of the choice range (3.5, 4.0, 4.5, 5.0, 5.5). Choice (B) is the only one consistent with 3.5π very close to 11.

💡Key takeaway

This AMC 8 problem only needs Grade 7 circle-area know-how plus the "break the shape into pieces and subtract" idea you already use for L-shaped figures.

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