Draw Pentagon From Circle Phi

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Regular Pentagon

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The regular pentagon is the regular polygon with v sides, equally illustrated above.

A number of distance relationships between vertices of the regular pentagon can be derived by similar triangles in the higher up left figure,

(one)

where is the diagonal distance. But the dashed vertical line connecting 2 nonadjacent polygon vertices is the same length equally the diagonal i, so

(ii)

(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

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

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

Five regular pentagons tin 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 ane, the outset ring of pentagons has centers at radius , the second ring at , and the th at .

In proposition 4.11, Euclid showed how to inscribe a regular pentagon in a circle. Ptolemy also gave a ruler and compass structure for the pentagon in his epoch-making work The Almagest. While Ptolemy'south construction has a simplicity of 16, a geometric structure using Carlyle circles can be made with geometrography symbol , which has simplicity 15 (DeTemple 1991).

The following elegant structure for the regular pentagon is due to Richmond (1893). Given a bespeak, 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 bore . The diameter perpendicular to the original diameter may exist synthetic by finding the perpendicular bisector. Call the upper endpoint of this perpendicular diameter . For the pentagon, detect the midpoint of and call information technology . Describe , and bisect , calling the intersection signal 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 past folding and knotting a strip of paper.

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See as well

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, Due west. W. R. and Coxeter, H. S. 1000. 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, Fifty. E. "Regular Pentagon and Decagon." §viii.17 in Monographs on Topics of Mod Mathematics Relevant to the Elementary Field (Ed. J. West. A. Young). New York: Dover, pp. 368-370, 1955. Dixon, R. Mathographics. New York: Dover, p. 17, 1991. Dudeney, H. East. 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, Thousand. "A Simple Compass-Only Structure of the Regular Pentagon." Forum Geom. 8, 147-148, 2008. Madachy, J. S. Madachy'due south Mathematical Recreations. New York: Dover, p. 59, 1979. Pappas, T. "The Pentagon, the Pentagram & the Gilded Triangle." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 188-189, 1989. Richmond, H. West. "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. i, 366-372, 1836. Wells, D. The Penguin Lexicon of Curious and Interesting Geometry. London: Penguin, p. 211, 1991.

Cite this as:

Weisstein, Eric Due west. "Regular Pentagon." From MathWorld--A Wolfram Spider web Resource. https://mathworld.wolfram.com/RegularPentagon.html

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