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The diagram below shows a regular hexagon with side length 1, inscribed in a square. Two of the vertices lie on the diagonal of the square and the remaining vertices lie on its sides. What is the area of the square?
The two vertices on the square diagonal are opposite vertices of the regular hexagon, so the hexagon and square have the same centre. Since a unit side of the hexagon makes an angle of $$15^\circ$$ with a square side, the width of the hexagon is $$2\cos15^\circ$$. Thus the square side is $$2\cos15^\circ$$ and its area is $$4\cos^2 15^\circ=2+\sqrt{3}$$.
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