Problem 1 · 2022 AMC 8
Easy
Geometry & Measurement
area-decompositiongrid-counting
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Answer: A — 10 square inches.
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Hint 1 of 2
The shape is slanted, but the grid behind it isn't — don't measure tilted lengths, let the grid count the area for you.
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Hint 2 of 2
Slice the gray region along the grid lines: every piece becomes a whole unit square (area 1) or a half-square triangle (area ½). Count each kind and add.
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Approach: let the grid count the area for you (whole squares + half-squares)
- Insight: a slanted shape on a grid is hard to eyeball, but it's easy to count. Slice the gray region along the grid lines and every piece becomes either a whole unit square or a triangular half-square — areas you know instantly (1 and ½).
- There are 4 whole gray squares in the middle; the surrounding gray triangles are twelve half-squares, adding 12 × ½ = 6 more.
- Total area: 4 + 6 = 10 square inches.
- You'll see this again: whenever a tilted figure sits on a grid, don't reach for slant-length formulas — chop it into unit squares and half-squares and just add.
Another way — spot five identical tilted unit-squares (MAA):
- The logo is made of five small tilted squares whose sides go diagonally across one grid cell, so each side is √(12 + 12) = √2.
- Each such square has area (√2)2 = 2.
- Five of them: 5 × 2 = 10 square inches.
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