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)
|
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)
|
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 diagonals shown in the construction above can also be used to locate the centers and the points where the four circular arcs of the pentagon egg meet.
The coordinates of the polygon vertices of a regular pentagon inscribed in a unit circle relative to the center of the pentagon are shown in the figure above, 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)
|
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.
