Problem 4 · 2018 AMC 8
Easy
Geometry & Measurement
area-decompositionarea
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Answer: C — 13 sq cm.
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Hint 1 of 2
Don't try to measure the whole jagged outline at once. Look for the "calm" piece in the middle — there's a plain square hiding in there, and the rest is just four matching points sticking out.
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Hint 2 of 2
The technique for any weird grid shape: cut it into pieces you already know (squares and right triangles), find each area, and add. The symmetry here means you only compute one triangle and multiply by four.
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Approach: split into a square + 4 triangles
- Spot the structure: a calm 3 × 3 square in the center, with one identical triangular point poking out of each of its four sides.
- Square area: 3 × 3 = 9. Each point is a triangle with base 2 and height 1, so area (1/2)(2)(1) = 1; four of them give 4.
- Total: 9 + 4 = 13 cm2. Sanity check: the figure is clearly bigger than the 9 square but doesn't fill its 5×5 bounding box, so 13 feels right.
- You'll see it again: spotting a symmetric core plus repeated identical flaps turns a 12-sided monster into "one square + 4 copies of one triangle."
Another way — Pick's Theorem (lattice points):
- Here's a power tool for any polygon whose corners sit on grid points: area = (interior dots) + (boundary dots)/2 − 1.
- Count the grid dots strictly inside the figure (the interior count) and the dots lying on its outline, plug into the formula, and you get 13 — no slicing into triangles needed. Worth knowing for any "shape drawn on graph paper" question.
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