AMC 10 · 2007 · #15

Grade 8 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Pythagorean / Distance Formula for a Length
tangent-circlespythagorean-theoremarea-rectangles convert-to-algebra ↑ Prerequisites: pythagorean-theorem
📏 Medium solution 💡 2 insights 📊 Diagram
Problem
A square holds four circles of radius 1, each tucked into a corner so it touches two sides of the square. A bigger circle of radius 2 sits in the middle and just touches each of the four small circles from the outside. Find the area of the square.

Pick an answer.

(A)
32
(B)
$22 + 12\sqrt {2}$
(C)
$16 + 16\sqrt {3}$
(D)
48
(E)
$36 + 16\sqrt {2}$

AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

This is a placement puzzle, so Tool #1 (Draw a Diagram) with coordinates pins down where every center sits. Put the shared center of the square and the big circle at the origin; then a corner circle's center is forced by the fact that it must stay one radius from each side it touches. Tool #7 (Identify Subproblems) isolates the one number that unlocks everything: the distance from the origin to a corner circle's center, which the external-tangency condition fixes at 2+1=3. Tool #4 (Introduce a Variable) names the side length s, turns that distance into an equation, and solves it; squaring gives the area.

1STEP 1

Set up coordinates

Put the shared center at the origin and let s be the side. A corner circle sits 1 in from each side, so its center is (s/2-1, s/2-1).

corner-circle center = (s/2-1, s/2-1)
2STEP 2

Read the distance from tangency

Externally tangent circles are apart by the sum of their radii, so the origin is 2 + 1 = 3 from that corner center.

distance from origin to center = 2 + 1 = 3
3STEP 3

Build and solve the equation

With a = s/2-1, the point (a,a) is a√(2) from the origin, so a√(2) = 3, a = 3√(2)/2, and s = 2 + 3√(2).

a√(2) = 3 → a = 3√(2)/2 → s = 2(1 + 3√(2)/2) = 2 + 3√(2)
4STEP 4

Square the side to get the area

Squaring the side: s² = (2 + 3√(2))² = 4 + 12√(2) + 18 = 22 + 12√(2), which is choice (B).

s² = (2 + 3√(2))² = 4 + 12√(2) + 18 = 22 + 12√(2) → (B)
Answer
22 + 12√(2)
Estimate to make sure the answer is sane. Since √(2) ≈ 1.414, the side is s ≈ 2 + 3(1.414) = 2 + 4.24 = 6.24, so the area is about 6.24² ≈ 39. Choice (B) gives 22 + 12(1.414) ≈ 22 + 17 = 39, which matches. It also passes a sanity floor: the big circle alone has diameter 4 across the middle, and the small circles add margin on each side, so a side near 6 and an area near 39 is exactly what the picture demands. Choices (A) 32 and (D) 48 have no radical and are too clean for this slanted setup; (C) uses √(3), which never appears here; (E) ≈ 59 is far too big.
💡Key takeaway

Pin every center on a grid, let each tangency hand you a distance, and one Pythagorean step turns those distances into the square's side.

  • Set up coordinates
  • Read the distance from tangency
  • Build and solve the equation
  • Square the side to get the area

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