AMC 10 · 2005 · #7

Grade 8 geometry-2d
Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps
area-circlesarea-rectanglesratio-proportionpythagorean-theorem ratio-proportionidentify-subproblems ↑ Prerequisites: area-circlesarea-rectanglespythagorean-theorem
📏 Medium solution 💡 2 insights
Problem
Start with a square. Inscribe a circle in it, then inscribe a square in that circle, then inscribe a circle in that last square. That final circle is the smallest circle and the first square is the largest square. Find the ratio of the smallest circle's area to the largest square's area.

Pick an answer.

(A)
$\frac{\pi}{16}$
(B)
$\frac{\pi}{8}$
(C)
$\frac{3\pi}{16}$
(D)
$\frac{\pi}{4}$
(E)
$\frac{\pi}{2}$

AMC 10 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

The figure is a nest of shapes inside shapes, so Tool #1 (Draw a Diagram) is the natural way in — a clear picture shows exactly where each circle touches its square and where each square's corners land. Because nothing gives an actual size, I fix the largest square's side to a convenient number and let the shapes cascade inward. Tool #7 (Identify Subproblems) handles the cascade: peel the nest one layer at a time, carrying a single length from each shape to the next. At the end Tool #3 (Eliminate Possibilities) matches my computed ratio to one of the five answer choices.

1STEP 1

Size the outer square and its circle

Let the largest square have side 2, so its area is 4; the inscribed circle spans one full side, giving diameter 2 and radius 1.

largest square area = 2² = 4, inner circle diameter = 2, radius = 1
2STEP 2

Middle square from the circle's diagonal

Its four corners sit on that circle, so its diagonal is the diameter 2; from x² + x² = 2² we get x² = 2 and side x = √2.

x² + x² = 2² → 2x² = 4 → x² = 2 → x = √(2)
3STEP 3

Smallest circle and its area

It is inscribed in that middle square, so its diameter is √2 and its radius √2/2; the area is π(√2/2)² = π/2.

radius = √(2)/2, area = π(√(2)/2)² = π·2/4 = π/2
4STEP 4

Form the ratio and match a choice

Divide the two areas: (π/2)/4 = π/8, which matches answer choice (B).

(smallest circle)/(largest square) = (π/2)/4 = π/8 → (B)
Answer
π/8
The smallest circle sits deep inside the largest square, so it should cover well under half of it — and π/8 ≈ 0.39, comfortably less than half, which fits the picture. The choice of side 2 was only for convenience: with a general side s the largest square has area s² and the smallest circle has area (π s²)/8, so the s² cancels and the ratio is π/8 no matter the starting size. Trap answer π/16 comes from going one shape too far inward, and π/4 from stopping one shape too early.
💡Key takeaway

Carry one length from each shape to the next: an inscribed circle's diameter equals its square's side, and an inscribed square's diagonal equals its circle's diameter — chain those and the sizes fall out.

  • Size the outer square and its circle
  • Middle square from the circle's diagonal
  • Smallest circle and its area
  • Form the ratio and match a choice

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