Theorem 5.8a: The area of an {n/k} star inscribed in a circle of radius r is

n[r2cos2(180k/n)sec2(180(k-1)/n)sin(180/n)cos(180/n) + s2sin(180k/n)cos(180k/n)]

proof: Consider the following figure.

We can divide the star into n quadrilateral regions formed by putting the red and green triangles together, as shown above. The red and green triangle are both isosceles triangles. Since rk-1 = rcos(180k/n)sec(180(k-1)/n) by Theorem 4.3, the area of the green triangle is r2cos2(180k/n)sec2(180(k-1)/n)sin(180/n)cos(180/n) by Corollary 5.1b. The area of the red triangle is s2sin(180k/n)cos(180k/n) also by Corollary 5.1b.

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