Theorem 5.5a: The area of a {n/k} star is

proof 2: Since we know from Theorem 4.3 that the distance from O to P1 is rcos(180k/n)sec(180/n), we could use Corollary 5.1b to get the area of the triangle to be

 

r2cos2(180k/n)sec2(180/n)sin(180/n)cos(180/n)

 

= r2cos2(180k/n)tan(180/n)

The summation term is the sum of the triangles and quadrilaterals from Theorem 5.4a.

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proof 1