Problem 15 · 2002 AMC 8
Hard
Geometry & Measurement
area-decompositiongrid
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Answer: E — Polygon E, with area 5.5.
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Hint 1 of 2
Slanted edges scare people off — but every tilted piece here is just *half of a grid square*. So you never measure a slant, you only count squares and half-squares.
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Hint 2 of 2
Tally inside each shape: a full grid square is worth 1, a corner-to-corner triangle is worth ½. Add them up per polygon and compare the totals.
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Approach: count whole squares and half-squares
- On a grid you don't need any area formulas: chop each polygon into pieces you can count — whole unit squares (area 1) and half-square triangles (area ½) — then add.
- Doing that for all five gives areas 5, 5, 5, 4.5, and 5.5. The biggest is polygon E at 5.5.
- *The transferable move:* break a messy shape into easy standard pieces and sum them. Counting squares-and-half-squares turns any grid polygon into simple addition.
Another way — box-and-subtract:
- Trap each polygon in the smallest grid rectangle that surrounds it, then subtract the right-triangle corners poking outside the polygon.
- Rectangle area minus the cut-off corners gives the same totals — handy when a shape has few inside squares but many slanted edges.
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