The regular pentagon is the regular polygon with five sides, as illustrated above.
A number of distance relationships between polygon 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 length of a polygon diagonal. But the
dashed vertical line connecting two nonadjacent polygon
vertices is the same length as this polygon diagonal,
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 polygon 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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These coordinates are the real parts and imaginary parts of the fifth roots
of unity, namely 
,
, and 
. They are the five roots
of 
.
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)
|
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 geometric construction for the pentagon
in his epoch-making work The Almagest. While Ptolemy's geometric
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 geometric 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.
