Competition · AMC preparation · step 4 of 4

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)
1
(C)
$\frac{3}2$
(D)
2
(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

Read the lengths from the figure

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

Find the large square's area

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

A_large = 2² = 4
3STEP 3

Find the small square's area

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

s√(2) = 2 → s² = 2²/2 = 2, A_small = s² = 2
4STEP 4

Find the circle's area

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

Build the two shaded regions

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

Estimate the ratio

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

(π - 2)/2 ≈ (3.14 - 2)/2 = 1.14/2 = 0.57
7STEP 7

Pick the closest choice

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 12\frac{1}{2}.

(A) 0.5, (B) 1, (C) 1.5, (D) 2, (E) 2.5 → (A) 1/2
Answer
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 12\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.

  • Read the lengths from the figure
  • Find the large square's area
  • Find the small square's area
  • Find the circle's area
  • Build the two shaded regions
  • Estimate the ratio
  • Pick the closest choice

A parent dashboard for the family lives at sensimlab.com.

More like this

Same problem type. Tags show what each one shares with this problem.