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Geometry Difficulty 4.6 AIME Prove it South Africa

An equilateral triangle and a regular hexagon have equal perimeters. What is the ratio of the area of the triangle to the area of the hexagon?

Solution

Let the perimeter be $6k$ . The equilateral triangle will have side length $2k$ , and will have an area four times the area of a triangle with side length $k$ .

The hexagon will have side length $k$ . By joining the vertices of the hexagon to the centre, it can be divided exactly into six equilateral triangles of side length $k$ .

The ratio of the area of the triangle to the area of the hexagon is then $4:6 = 2:3$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.