Theorem 5.8c: The area of an {n/k} star inscribed in a
circle of radius r is
nr2sin(180/n)cos(180k/n)sec(180(k-1)/n)
proof 1: We start with the formula from
Theorem 5.8a
n[r2cos2(180k/n)sec2(180(k-1)/n)sin(180/n)cos(180/n)
+ s2sin(180k/n)cos(180k/n)]
Substitute for s = rsin(180/n)sec(180(k-1)/n) from
Theorem 5.7c, and our area
becomes
n[r2cos2(180k/n)sec2(180(k-1)/n)sin(180/n)cos(180/n)
+ r2sec2(180(k-1)/n)sin2(180/n)sin(
180k/n)cos(180k/n)]
=
nr2sin(180/n)cos(180k/n)sec2(180(k-1)/n)[cos(180k/n)cos(180/n)
+ sin(180k/n)sin(180/n)]
=
nr2sin(180/n)cos(180k/n)sec2(180(k-1)/n)cos(180k/n-180/n)
=
nr2sin(180/n)cos(180k/n)sec2(180(k-1)/n)cos(180(k-1)/n)
= nr2sin(180/n)cos(180k/n)sec(180(k-1)/n)
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Proof 2
Proof 3
Proof 4
Proof 5
Proof 6