Problem 23 · 2007 AMC 8
Hard
Geometry & Measurement
subtract-from-whole
What is the area of the shaded pinwheel shown in the 5 × 5 grid?
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Answer: B — 6.
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Hint 1 of 2
The shaded pinwheel has slanted, awkward edges — but the unshaded leftovers are tidy squares and triangles. Measure what's easy and subtract from the whole 5×5.
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Hint 2 of 2
Complementary area: when the region you want is jagged, find the area of its simple complement and subtract from the total.
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Approach: whole grid minus the easy unshaded pieces
- Total grid area = 52 = 25.
- The unshaded part is 4 corner unit squares (4 · 1 = 4) plus 4 congruent triangles, each with base 3 and height 5/2.
- Those triangles total 4 · (1/2)(3)(5/2) = 15. Unshaded = 4 + 15 = 19.
- Shaded pinwheel = 25 − 19 = 6.
- Reusable move: never wrestle a slanted shape directly — subtracting the clean complement from a bounding rectangle is almost always faster and less error-prone.
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