Maths Olympiad Prep

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, 2018

Geometry Difficulty 4.2 AIME Find the answer United States

Problem:

An equilateral hexagon with side length 11 has interior angles 90∘,120∘,150∘,90∘,120∘,150∘90^{\circ}, 120^{\circ}, 150^{\circ}, 90^{\circ}, 120^{\circ}, 150^{\circ} in that order. Find its area.

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}

The area of this hexagon is the area of a 32×(1+32)\frac{3}{2} \times \left(1+\frac{\sqrt{3}}{2}\right) rectangle (with the 90∘90^{\circ} angles of the hexagon at opposite vertices) minus the area of an equilateral triangle with side length 11. Then this is
6+334−34=3+32 \frac{6+3 \sqrt{3}}{4} - \frac{\sqrt{3}}{4} = \frac{3+\sqrt{3}}{2}

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