Maths Olympiad Prep

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Geometry Difficulty 4.3 AIME Find the answer United States

Problem:

Consider an equilateral triangle and a square both inscribed in a unit circle such that one side of the square is parallel to one side of the triangle. Compute the area of the convex heptagon formed by the vertices of both the triangle and the square.

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:

Answer: 3+32\frac{3+\sqrt{3}}{2}

Figure 1

Consider the diagram above. We see that the shape is a square plus 3 triangles. The top and bottom triangles have base 2\sqrt{2} and height 12(3−2)\frac{1}{2}(\sqrt{3}-\sqrt{2}), and the triangle on the side has the same base and height 1−221-\frac{\sqrt{2}}{2}. Adding their areas, we get the answer.

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