This puzzle isn't too hard, but it will introduce you to a cool geometrical object -- the Reuleaux triangle (pronounced "Roo-low"). This "triangle" is pictured in the diagram at right (the yellow shape). As you can see, it really isn't a triangle at all because its sides aren't straight. To build a Reuleaux triangle, start with the three vertices of an equilateral triangle (with all three sides of length 1). Now draw three circles. Each circle is centered at one of the three vertices, and passes through the other two vertices. The area of overlap between these circular disks is the Reuleaux triangle.

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This shape has several interesting properties. One of them is that it is the same width no matter which way it is rotated. (That is, it will always just barely fit through an opening 1 unit wide). The circle, of course, has this same property, but not many non-circular shapes do. A square (with side-length 1), for instance, will fit through an opening of width 1 if taken through with sides parallel to the opening. But otherwise, it won't fit!

Have you ever wondered why manhole covers are round instead of square? The answer is found in this very property: a square manhole cover could be picked up, rotated, and dropped through the hole it covers. However, a circular manhole cover won't fit through that hole no matter how it is rotated. Maybe sometime someone will manufacture manhole covers shaped like the Reuleaux triangle!

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Now for the puzzle: find the exact area of the Reuleaux triangle! Be sure to simplify your answer as much as possible.

Good luck!