A_{tot} = A [\frac{sin(Nϕ/2)}{sin(ϕ/2)}]

L = R sin (\frac{α}{2})

A_{tot} = 2L = 2R sin (\frac{α}{2})

M = R sin (\frac{ϕ}{2})

A = 2M = 2R sin (\frac{ϕ}{2})

\frac{A_{tot}}{A} = \frac{2R sin (\frac{α}{2})}{2R sin (\frac{ϕ}{2})}

α = N ϕ

\frac{A_{tot}}{A} = \frac{2R sin (\frac{Nϕ}{2})}{2R sin (\frac{ϕ}{2})}

A_{tot} = A [ \begin{matrix}sin & (\frac{Nϕ}{2}) \\ sin & (\frac{ϕ}{2}) \end{matrix}]