Find area of a seven-point star inscribed in a circle of radius 1

Question

I need to know area of the 7 point star and radius of the circle is 1.

Perhaps here are some useful information and formulas: https://en.wikipedia.org/wiki/Prism_(geometry) — Matti P., Jan 21 '20 at 13:12

score 2 · Answer 1 · answered Jan 21 '20 at 16:01

Note that $\angle BOA = \frac\pi7$ and $\angle OAB = \frac\pi{14}$. Apply the sine rule to the triangle AOB,

$$\frac{OB}{OA} = \frac{\sin\frac\pi{14}}{\sin \frac{3\pi}{14}} =\frac{1}{3-4\sin^2 \frac{\pi}{14}}=\frac{1}{1+2\cos \frac{\pi}{7}}$$

Thus, with the radius AO = 1, the area is

$$A= 14A_{AOB} = 14\cdot \frac12 AO\cdot OB \sin\frac\pi7=\frac{7\sin\frac\pi7}{1+2\cos \frac{\pi}{7}}$$

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