 TOPICS # Regular Pentagon The regular pentagon is the regular polygon with five sides, as illustrated above. A number of distance relationships between vertices of the regular pentagon can be derived by similar triangles in the above left figure, (1)

where is the diagonal distance. But the dashed vertical line connecting two nonadjacent polygon vertices is the same length as the diagonal one, so (2) (3)

Solving the quadratic equation and taking the plus sign (since the distance must be positive) gives the golden ratio (4) The coordinates of the vertices of a regular pentagon inscribed in a unit circle relative to the center of the pentagon are given as shown in the above figures, with   (5)   (6)   (7)   (8)

The circumradius, inradius, sagitta, and area of a regular pentagon of side length are given by   (9)   (10)   (11)   (12)   (13)   (14)     (15)

where is the golden ratio. The height of a regular pentagon of side length is given by   (16)     (17)

Five regular pentagons can be arranged around an identical pentagon to form the first iteration of the "pentaflake," which itself has the shape of a regular 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 (DeTemple 1991). The following elegant construction for the regular 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 , and copying the angle then gives the remaining points , , and (Coxeter 1969, Wells 1991).

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

Associahedron, Cyclic Pentagon, Decagon, Dissection, Five Disks Problem, Home Plate, Pentaflake, Pentagon, Pentagram, Polygon, Regular Polygon, Trigonometry Angles--Pi/5

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## 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.DeTemple, D. W. "Carlyle Circles and the Lemoine Simplicity of Polygonal Constructions." Amer. Math. Monthly 98, 97-108, 1991.Dickson, L. E. "Regular Pentagon and Decagon." §8.17 in Monographs on Topics of Modern Mathematics Relevant to the Elementary Field (Ed. J. W. A. Young). New York: Dover, pp. 368-370, 1955.Dixon, R. Mathographics. New York: Dover, p. 17, 1991.Dudeney, H. E. Amusements in Mathematics. New York: Dover, p. 38, 1970.Fukagawa, H. and Pedoe, D. "Pentagons." §4.3 in Japanese Temple Geometry Problems. Winnipeg, Manitoba, Canada: Charles Babbage Research Foundation, pp. 49 and 132-134, 1989.Hofstetter, K. "A Simple Compass-Only Construction of the Regular Pentagon." Forum Geom. 8, 147-148, 2008.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.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 211, 1991.

## Cite this as:

Weisstein, Eric W. "Regular Pentagon." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RegularPentagon.html