AMC 10 · 2005 · #5

Grade 7 geometry-2d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Circle-Polygon Hybrid
area-circlesarea-rectanglesarea-difference identify-subproblemsarea-differencespatial-visualization ↑ Prerequisites: area-circlesarea-rectangles
📏 Medium solution 💡 2 insights 📊 Diagram
Problem
A floor 8 by 10 feet is covered by unit square tiles. On each tile a white quarter circle of radius one half sits at every corner, and the rest is shaded. Find the total shaded area.

Pick an answer.

(A)
$80-20\pi$
(B)
$60-10\pi$
(C)
$80-10\pi$
(D)
$60+10\pi$
(E)
$80+10\pi$
How to solve
Strategy Identify Subproblems

The whole floor is too big to attack at once, so Tool #7 (Identify Subproblems) splits it into two easy questions: how many tiles are there, and how much of one tile is shaded? Tool #17 (Visualize Spatial Relationships) is the key unlock on a single tile — the four corner quarter circles can be imagined sliding together into one whole circle, turning an awkward four-piece white region into a single circle whose area is easy. Tool #16 (Count the Complement) then gets the shaded part the short way: instead of measuring the strange shaded shape directly, take the tile area and subtract the white circle. Multiply the per-tile shaded amount by the tile count to finish.

1STEP 1

Count the tiles

The floor holds 80 identical tiles.

8×10 = 80 tiles
2STEP 2

Slide the quarters into one circle

The four corner pieces slide together into one circle.

4×1/4 circle = 1 full circle of radius 1/2
3STEP 3

Find the white area on a tile

That circle has area π/4.

π r² = π(1/2)² = π/4
4STEP 4

Shaded area on one tile

Subtracting gives the shaded part of one tile.

1 - π/4
5STEP 5

Scale up to the whole floor

Scaling up gives 80-20π, choice (A).

80(1-π/4) = 80-20π → (A)
Answer
80-20π
Check the size: π/4≈0.785, so each tile is about 1-0.785=0.215 square feet shaded, and 80×0.215≈17.2 square feet total. The formula gives 80-20π≈80-62.8=17.2 — a match. The value is a positive number less than 80, which makes sense because the shaded part must be smaller than the whole floor. The '+' choices (D) and (E) are bigger than 80, impossible for a part of an 80-square-foot floor, so they can be dropped immediately. Choice (C) 80-10π comes from forgetting that a full circle, not a half circle, is subtracted per tile; choice (B) 60-10π mis-sizes both the floor and the circle.
💡Key takeaway

Four corner quarter circles make one whole circle, so each tile loses π/4 to white; across all 80 tiles the shaded floor is 80-20π square feet.

  • Count the tiles
  • Slide the quarters into one circle
  • Find the white area on a tile
  • Shaded area on one tile
  • Scale up to the whole floor

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