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 2: We start with the formula from
Theorem 5.8b
n[r2sin(180/n)cos(180/n) -
s2sin(180(k-1)/n)cos(180(k-1)/n)]
and substitute s = rsec(180(k-1)/n)sin(180/n), from
Theorem 5.7c, to get
= n[r2sin(180/n)cos(180/n) -
r2sec2(180(k-1)/n)sin2(180/n)sin(180(k-1)/n)cos(180(k-1)/n)]
= n[r2sin(180/n)cos(180/n) -
r2sec(180(k-1)/n)sin2(180/n)sin(180(k-1)/n)]
= nr2sin(180/n)[cos(180/n) -
sec(180(k-1)/n)sin(180/n)sin(180(k-1)/n) ]
= nr2sin(180/n)cos(180/n +
180(k-1)/n)sec(180(k-1)/n)
= nr2sin(180/n)cos(180k/n)sec(180(k-1)/n)
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Proof 1
Proof 3
Proof 4
Proof 5
Proof 6