The regular convex 5-gon is called the pentagon. By Similar Triangles in the figure on the
left,

(1) |

(2) |

(3) |

(4) |

(5) |

The coordinates of the Vertices relative to the center of the pentagon with unit sides are given as
shown in the above figure, with

(6) | |||

(7) | |||

(8) | |||

(9) |

For a regular Polygon, the Circumradius, Inradius, Sagitta, and Area are given by

(10) | |||

(11) | |||

(12) | |||

(13) |

Plugging in gives

(14) | |||

(15) | |||

(16) | |||

(17) |

Five pentagons can be arranged around an identical pentagon to form the first iteration of the ``Pentaflake,'' which itself has the shape of a pentagon with five triangular wedges removed. For a pentagon of side length 1, the first ring of pentagons has centers at radius , the second ring at , and the th at .

In proposition IV.11, Euclid showed how to inscribe a regular pentagon in a Circle. Ptolemy
also gave a Ruler and Compass construction for the pentagon in his epoch-making work *The Almagest.*
While Ptolemy's construction has a Simplicity of 16, a Geometric Construction using Carlyle
Circles can be made with Geometrography symbol
, which has
Simplicity 15 (De Temple 1991).

The following elegant construction for the pentagon is due to Richmond (1893). Given a point, a Circle may be constructed of any desired Radius, and a Diameter drawn through the center. Call the center , and the right end of the Diameter . The Diameter Perpendicular to the original Diameter may be constructed by finding the Perpendicular Bisector. Call the upper endpoint of this Perpendicular Diameter . For the pentagon, find the Midpoint of and call it . Draw , and Bisect , calling the intersection point with . Draw Parallel to , and the first two points of the pentagon are and (Coxeter 1969).

Madachy (1979) illustrates how to construct a pentagon by folding and knotting a strip of paper.

**References**

Ball, W. W. R. and Coxeter, H. S. M. *Mathematical Recreations and Essays, 13th ed.*
New York: Dover, pp. 95-96, 1987.

Coxeter, H. S. M. *Introduction to Geometry, 2nd ed.* New York: Wiley, pp. 26-28, 1969.

De Temple, D. W. ``Carlyle Circles and the Lemoine Simplicity of Polygonal Constructions.'' *Amer. Math. Monthly*
**98**, 97-108, 1991.

Dixon, R. *Mathographics.* New York: Dover, p. 17, 1991.

Dudeney, H. E. *Amusements in Mathematics.* New York: Dover, p. 38, 1970.

Madachy, J. S. *Madachy's Mathematical Recreations.* New York: Dover, p. 59, 1979.

Pappas, T. ``The Pentagon, the Pentagram & the Golden Triangle.'' *The Joy of Mathematics.*
San Carlos, CA: Wide World Publ./Tetra, pp. 188-189, 1989.

Richmond, H. W. ``A Construction for a Regular Polygon of Seventeen Sides.'' *Quart. J. Pure Appl. Math.* **26**, 206-207, 1893.

Wantzel, M. L. ``Recherches sur les moyens de reconnaître si un Problème de Géométrie peut se
résoudre avec la règle et le compas.'' *J. Math. pures appliq.* **1**, 366-372, 1836.

© 1996-9

1999-05-26