Problem 12 · 2007 AMC 8
Medium
Geometry & Measurement
hexagon-decomposition
A unit hexagram is composed of a regular hexagon of side length 1 and its 6 equilateral triangular extensions, as shown in the diagram. What is the ratio of the area of the extensions to the area of the original hexagon?
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Answer: A — 1 : 1.
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Hint 1 of 2
Don't compute any area — count triangles. Draw the three long diagonals of the hexagon and it falls into 6 little triangles that are identical to the 6 spikes glued on the outside.
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Hint 2 of 2
Match shapes instead of measuring: if you can show two regions are made of congruent pieces, their areas are equal — no formula required.
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Approach: cut the hexagon into copies of the spikes
- Connect the hexagon's center to its 6 corners. A regular hexagon splits into 6 equilateral triangles, each with side 1.
- Each outer spike is also an equilateral triangle of side 1, built on one edge of the hexagon — so it's congruent to one of the inner triangles.
- 6 inner triangles vs. 6 identical outer triangles: the areas are equal, so the ratio is 1 : 1.
- Reusable idea: a regular hexagon is exactly 6 equilateral triangles of its side length — a fact that turns most hexagon-area problems into simple counting.
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