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,


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


Solving the quadratic equation and taking the plus sign (since the distance must be positive) gives the golden ratio


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


The circumradius, inradius, sagitta, and area of a regular pentagon of side length a are given by


where phi is the golden ratio. The height of a regular pentagon of side length a is given by


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 phi, the second ring at phi^3, and the nth at phi^(2n-1).


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 2S_1+S_2+8C_1+0C_2+4C_3, 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 O, and the right end of the diameter P_1. The diameter perpendicular to the original diameter may be constructed by finding the perpendicular bisector. Call the upper endpoint of this perpendicular diameter B. For the pentagon, find the midpoint of OB and call it D. Draw DP_1, and bisect ∠ODP_1, calling the intersection point with OP_1 N_2. Draw N_2P_2 parallel to OB, and the first two points of the pentagon are P_1 and P_2, and copying the angle ∠P_1OP_2 then gives the remaining points P_3, P_4, and P_5 (Coxeter 1969, Wells 1991).

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

See also

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

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Cite this as:

Weisstein, Eric W. "Regular Pentagon." From MathWorld--A Wolfram Web Resource.

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