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mathisfun1 · #5 July 19th, 2007, 07:36 AM

From law of cosines, each segment connecting two adjacent points on the perimeter have length $2\sin{\frac{\pi}{7}}$ . Each of the next-longest segments has length $2\sin{\frac{2\pi}{7}}$ , and a longest segment has length $2\sin{\frac{3\pi}{7}}$ . There are seven segments of each length, so the product is $(8\sin{\frac{\pi}{7}}\sin{\frac{2\pi}{7}}\sin{\frac{3\pi}{7}})^7$ . Using the well-known fact that $\sin{\frac{\pi}{7}}\sin{\frac{2\pi}{7}}\sin{\frac{3\pi}{7}}=\frac{\sqrt 7}{8}$ Trigonometry Angles--Pi/7 -- from Wolfram MathWorld, our answer is $243\sqrt{7}$ .