In Euclidean geometry, a regular polygon is
a polygon that is equiangular and equilateral.
Regular polygons may be convex or star. In
the limit, a sequence of regular polygons
with an increasing number of sides becomes
a circle, if the perimeter is fixed, or a
regular apeirogon, if the edge length is fixed.
General properties
These properties apply to all regular polygons,
whether convex or star.
A regular n-sided polygon has rotational symmetry
of order n.
All vertices of a regular polygon lie on a
common circle, i.e., they are concyclic points.
That is, a regular polygon is a cyclic polygon.
Together with the property of equal-length
sides, this implies that every regular polygon
also has an inscribed circle or incircle that
is tangent to every side at the midpoint.
Thus a regular polygon is a tangential polygon.
A regular n-sided polygon can be constructed
with compass and straightedge if and only
if the odd prime factors of n are distinct
Fermat primes. See constructible polygon.
Symmetry
The symmetry group of an n-sided regular polygon
is dihedral group Dn: D2, D3, D4, ... It consists
of the rotations in Cn, together with reflection
symmetry in n axes that pass through the center.
If n is even then half of these axes pass
through two opposite vertices, and the other
half through the midpoint of opposite sides.
If n is odd then all axes pass through a vertex
and the midpoint of the opposite side.
Regular convex polygons
All regular simple polygons are convex. Those
having the same number of sides are also similar.
An n-sided convex regular polygon is denoted
by its Schläfli symbol {n}.
Henagon or monogon {1}: degenerate in ordinary
space.
Digon {2}: a "double line segment": degenerate
in ordinary space.
In certain contexts all the polygons considered
will be regular. In such circumstances it
is customary to drop the prefix regular. For
instance, all the faces of uniform polyhedra
must be regular and the faces will be described
simply as triangle, square, pentagon, etc.
Angles
For a regular convex n-gon, each interior
angle has a measure of:
degrees,
or radians,
or full turns,
and each exterior angle has a measure of degrees,
with the sum of the exterior angles equal
to 360 degrees or 2π radians or one full
turn.
Diagonals
For n > 2 the number of diagonals is , i.e.,
0, 2, 5, 9, ... for a triangle, quadrilateral,
pentagon, hexagon, .... The diagonals divide
the polygon into 1, 4, 11, 24, ... pieces.
For a regular n-gon inscribed in a unit-radius
circle, the product of the distances from
a given vertex to all other vertices equals
n.
Interior points
For a regular n-gon, the sum of the perpendicular
distances from any interior point to the n
sides is n times the apothem. This is a generalization
of Viviani's theorem for the n=3 case.
Circumradius
The circumradius R from the center of a regular
polygon to one of the vertices is related
to the side length s or to the apothem a by
The sum of the perpendiculars from a regular
n-gon's vertices to any line tangent to the
circumcircle equals n times the circumradius.
The sum of the squared distances from the
vertices of a regular n-gon to any point on
its circumcircle equals 2nR2 where R is the
circumradius.
The sum of the squared distances from the
midpoints of the sides of a regular n-gon
to any point on the circumcircle is 2nR2 —/4,
where a is the side length and R is the circumradius.
Area
The area A of a convex regular n-sided polygon
having side s, circumradius R, apothem a,
and perimeter p is given by
For regular polygons with side s=1, circumradius
R =1, or apothem a=1, this produces the following
table:
Of all n-gons with a given perimeter, the
one with the largest area is regular.
Regular skew polygons
A regular skew polygon in 3-space can be seen
as nonplanar paths zig-zagging between two
parallel planes, defined as the side-edges
of a uniform antiprism. All edges and internal
angles are equal.
More generally regular skew polygons can be
defined in n-space. Examples include the Petrie
polygons, polygonal paths of edges that divide
a regular polytope into two halves, and seen
as a regular polygon in orthogonal projection.
In the infinite limit regular skew polygons
become skew apeirogons.
Regular star polygons
A non-convex regular polygon is a regular
star polygon. The most common example is the
pentagram, which has the same vertices as
a pentagon, but connects alternating vertices.
For an n-sided star polygon, the Schläfli
symbol is modified to indicate the density
or "starriness" m of the polygon, as {n/m}.
If m is 2, for example, then every second
point is joined. If m is 3, then every third
point is joined. The boundary of the polygon
winds around the center m times.
The regular stars of up to 12 sides are:
Pentagram – {5/2}
Heptagram – {7/2} and {7/3}
Octagram – {8/3}
Enneagram – {9/2} and {9/4}
Decagram – {10/3}
Hendecagram – {11/2}, {11/3}, {11/4} and
{11/5}
Dodecagram – {12/5}
m and n must be co-prime, or the figure will
degenerate.
The degenerate regular stars of up to 12 sides
are:
Hexagram – {6/2}
Octagram – {8/2}
Enneagram – {9/3}
Decagram – {10/2} and {10/4}
Dodecagram – {12/2}, {12/3} and {12/4}
Depending on the precise derivation of the
Schläfli symbol, opinions differ as to the
nature of the degenerate figure. For example
{6/2} may be treated in either of two ways:
For much of the 20th century), we have commonly
taken the /2 to indicate joining each vertex
of a convex {6} to its near neighbors two
steps away, to obtain the regular compound
of two triangles, or hexagram.
Many modern geometers, such as Grünbaum,
regard this as incorrect. They take the /2
to indicate moving two places around the {6}
at each step, obtaining a "double-wound" triangle
that has two vertices superimposed at each
corner point and two edges along each line
segment. Not only does this fit in better
with modern theories of abstract polytopes,
but it also more closely copies the way in
which Poinsot created his star polygons – by
taking a single length of wire and bending
it at successive points through the same angle
until the figure closed.
Duality of regular polygons
All regular polygons are self-dual to congruency,
and for odd n they are self-dual to identity.
In addition, the regular star figures, being
composed of regular polygons, are also self-dual.
Regular polygons as faces of polyhedra
A uniform polyhedron has regular polygons
as faces, such that for every two vertices
there is an isometry mapping one into the
other.
A quasiregular polyhedron is a uniform polyhedron
which has just two kinds of face alternating
around each vertex.
A regular polyhedron is a uniform polyhedron
which has just one kind of face.
The remaining convex polyhedra with regular
faces are known as the Johnson solids.
A polyhedron having regular triangles as faces
is called a deltahedron.
See also
Tiling by regular polygons
Platonic solids
Apeirogon – An infinite-sided polygon can
also be regular, {∞}.
List of regular polytopes
Equilateral polygon
Carlyle circle
Notes
References
Coxeter, H.S.M.. "Regular Polytopes". Methuen
and Co. 
Grünbaum, B.; Are your polyhedra the same
as my polyhedra?, Discrete and comput. geom:
the Goodman-Pollack festschrift, Ed. Aronov
et al., Springer, pp. 461–488.
Poinsot, L.; Memoire sur les polygones et
polyèdres. J. de l'École Polytechnique 9,
pp. 16–48.
External links
Weisstein, Eric W., "Regular polygon", MathWorld.
Regular Polygon description With interactive
animation
Incircle of a Regular Polygon With interactive
animation
Area of a Regular Polygon Three different
formulae, with interactive animation
Renaissance artists' constructions of regular
polygons at Convergence
