AMC 8 · 2011 · #25

Grade 8 geometry-2d

Archetype Compound Area by Decomposition / Difference └ Sub-archetype Nested Frame / Concentric Difference

area-circlesarea-rectanglesestimation identify-subproblemsarea-difference ↑ Prerequisites: area-circlesarea-rectangles

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

A circle of radius 1 sits inside a larger square (touching all four sides) and outside a smaller square (whose four corners are on the circle). The picture has two shaded pieces: the part of the circle outside the small square, and the part of the large square outside the circle. We need the ratio (circle-shaded area) : (area between the two squares), and pick the answer choice closest to that ratio.

Pick an answer.

(A)

$\frac{1}2$

(B)

(C)

$\frac{3}2$

(D)

(E)

$\frac{5}2$

AMC 8 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The picture has three nested shapes — large square, circle, small square — so Tool #7 (Identify Subproblems) splits the work into three clean area calculations and a final ratio. Tool #1 (Draw a Diagram) is built into the problem's figure: reading off that the circle's diameter equals the large square's side and the small square's diagonal is what unlocks every dimension. With a clean numerical ratio in hand, Tool #3 (Eliminate Possibilities) finishes the job by comparing one decimal to the five answer choices.

1STEP 1

Circle inscribed in the big square → its side is the diameter, 2. Small square inscribed in the circle → its diagonal is also 2.

side_large = 2, diagonal_small = 2

2STEP 2

Subproblem 1 — the large square's area is side squared: 2² = 4.

A_large = 2² = 4

3STEP 3

Subproblem 2 — split the small square along its diagonal: s√2 = 2 gives s² = 2, which is its area.

s√(2) = 2 → s² =

\frac{2²}{2}

= 2, A_small = s² = 2

4STEP 4

Subproblem 3 — the circle's area is πr² with r = 1, so it is just π.

A_circle = π r² = π (1)² = π

5STEP 5

Circle's shaded part = circle − small square = π - 2; the between-squares region = large − small = 2.

circle shaded = π - 2, between squares = 4 - 2 = 2

6STEP 6

Form the ratio and estimate with π ≈ 3.14: $\frac{(π - 2)}{2}$ ≈ $\frac{(3.14 - 2)}{2}$ = 0.57.

\frac{(π - 2)}{2}

≈

\frac{(3.14 - 2)}{2}

\frac{1.14}{2}

= 0.57

7STEP 7

0.57 is closest to 0.5 (gap 0.07) versus 1 (gap 0.43), and farther from the rest, so the nearest choice is $\frac{1}{2}$ .

(A) 0.5, (B) 1, (C) 1.5, (D) 2, (E) 2.5 → (A)

\frac{1}{2}

Answer

\frac{1}{2}

Sanity check the size: the circle (π ≈ 3.14) is a bit bigger than the small square (2) but smaller than the large square (4), which matches the picture. The circle's shaded crescents add up to a thin sliver — about 1.14 — and the four corner pieces between the squares add up to 2. So the shaded circle area is about half the between-squares area, and

\frac{1}{2}

matches.

💡Key takeaway

Three nested shapes break into three Grade 3-8 area facts, and a π ≈ 3.14 estimate turns the ratio into a single decimal you can match to the closest answer choice.

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