Let and be the Perimeters of the Circumscribed and
Inscribed -gon and and the Perimeters of the
Circumscribed and Inscribed -gon. Then

(1) | |||

(2) |

The first follows from the fact that side lengths of the Polygons on a Circle of Radius are

(3) | |||

(4) |

so

(5) | |||

(6) |

But

(7) |

Using the identity

(8) |

(9) |

(10) |

(11) |

(12) |

**References**

Dörrie, H. *100 Great Problems of Elementary Mathematics: Their History and Solutions.* New York: Dover,
p. 186, 1965.

© 1996-9

1999-05-25