AMC 10 · 2021 · #15

Grade 10 geometry-2d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Polygon Triangulation
rotation-isometrythirty-sixty-ninety-trianglearea-trianglestransformations-composition symmetry-argumentidentify-subproblems ↑ Prerequisites: area-trianglesrotation-isometry
📏 Long solution 💡 4 insights 📊 Diagram
Problem
Three identical square sheets of side length 6 are stacked sharing the same centre. The middle sheet is turned thirty degrees and the top sheet sixty degrees. The region covered by at least one sheet is a twenty-four-sided polygon. Write its area as a whole number minus a whole number times a square root, then add the three whole numbers.

Pick an answer.

(A)
75
(B)
93
(C)
96
(D)
129
(E)
147
How to solve
Strategy Visualize Spatial Relationships

A 24-sided polygon looks like 24 separate measurements, but the turning angles hide a shortcut. A square is unchanged by a quarter turn, so turning the whole stack another 30° sends the 0° sheet to 30°, the 30° sheet to 60°, and the 60° sheet to 90°, which is the 0° sheet again. The picture lands exactly on itself. That means the polygon repeats every 30°, so joining the center to the 24 sides cuts it into 24 identical triangles. Then only one triangle has to be measured. Each of those triangles has its base on a side of some square, and the center sits 3 away from every square side, so the height is free and only the base length needs work.

1STEP 1

Pin the center and two distances

Pin the two distances from the centre.

d(O, side) = 6/2 = 3, d(O, corner) = √(3² + 3²) = 3√(2)
2STEP 2

The picture repeats every 30 degrees

The picture repeats every thirty degrees.

{0°, 30°, 60°} → {30°, 60°, 90°} = {30°, 60°, 0°}
3STEP 3

Cut the polygon into 24 equal triangles

Cut it into identical triangles.

[polygon] = 24 · 1/2 · s · 3 = 36s
4STEP 4

Find where the next square cuts the side

Find where the next square cuts the side.

∠ MOX = 60°/2 = 30°, MX = OM tan 30° = 3 · 1/√(3) = √(3)
5STEP 5

One triangle, then all 24

Measure one triangle, then take twenty-four.

s = 3 - √(3), [polygon] = 24 · 1/2(3-√(3))(3) = 36(3-√(3)) = 108 - 36√(3)
6STEP 6

Read off a, b, and c

Reading and adding the three numbers gives 147.

a = 108, b = 36, c = 3 → a + b + c = 147
Answer
147
Check the size: 108 - 36√(3) ≈ 108 - 62.35 = 45.65. One sheet alone covers 36, so the union must be more than 36, and it must be less than the area of the regular 12-gon whose corners are the 12 square corners at distance 3√(2), which is 3(3√(2))² = 54. The value 45.65 sits comfortably between 36 and 54. The pieces also fit: each polygon side is 3 - √(3) ≈ 1.27, and 24 of them make a perimeter of about 30.4, sensible for a nearly round shape of that size. Finally the two kinds of polygon vertices land where they should, the square corners at 3√(2) ≈ 4.24 from the center and the crossing points at √((√(3))² + 3²) = 2√(3) ≈ 3.46, alternating 12 and 12 to make 24. So a + b + c = 147, choice (E).
💡Key takeaway

When turning a picture by some angle leaves it looking the same, measure just one slice and multiply, because every other slice is a copy of it.

  • Pin the center and two distances
  • The picture repeats every 30 degrees
  • Cut the polygon into 24 equal triangles
  • Find where the next square cuts the side
  • One triangle, then all 24
  • Read off a, b, and c

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