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,
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(1)
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where 
is the diagonal distance. But the dashed vertical line connecting two nonadjacent
polygon vertices is the same length as the diagonal
one, so
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(2)
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(3)
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Solving the quadratic equation and taking the plus sign (since the distance must be positive) gives the golden ratio
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(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
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(5)
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(6)
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(7)
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(8)
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The circumradius, inradius, sagitta, and area of a regular
pentagon of side length 
are given by
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(9)
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(10)
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(11)
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(12)
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(13)
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(14)
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(15)
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where 
is the golden ratio. The height
of a regular pentagon of side length 
is given by
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(16)
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(17)
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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.
