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

proof 1: The summation gives us the total area of all of the quadrilateral and triangular figures. To see that the first term gives us the area of the central n-gon, consider the following figure.

The n-gon consists of n triangles whose base is an edge and whose vertex is O, the center of the figure. Half of the base is s₁ the distance from P_o to P₁ which we know from Theorem 4.4a to be rcos(180k/n)tan(180/n). The distance from O to Po we know from Theorem 4.2 to be f = rcos(180k/n). Hence the area of the n-gon is nr²cos²(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 2