A pentagon is a five-sided polygon. Commonly, the term "pentagon" is used to refer to the regular pentagon.
The regular pentagon is the regular
polygon with five sides illustrated above.
A number of distance relationships between vertices of the pentagon can be derived by similar triangles in the
above left figure,
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(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)
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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  are given by
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 (DeTemple
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  , and copying
the angle  then gives the remaining
points  ,  , and  (Coxeter 1969,
Wells 1991).
Madachy (1979) illustrates how to construct a pentagon by folding and knotting a strip of paper.
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.
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