regular polygon

regular polygon

A

regular polygon

is a

polygon

such that all of its sides are

congruent

and all of its

interior angles

are congruent; that is, a polygon which is both

equilateral

and

equiangular

. Note that all regular polygons are convex.

There is a regular polygon with

n

sides for every

n

>

2

. As

n

increases, they resemble a circle ever more closely, and in fact were the basis for some early attempts to approximate the value of

π

. The existence of many types of regular polygons is in contrast with the case for

regular polyhedra

, of which only five distinct types in

Euclidean geometry

.

Some regular polygons have special names. For example, in many

geometries

, a

regular triangle

is also known as an

equilateral triangle

and (less commonly) an

equiangular triangle

. Also, in Euclidean geometry, a

regular

quadrilateral

is also known as a square. For larger numbers of sides, one uses a Greek number prefix, as in “

regular pentagon

” and “

regular hexagon

”. For numbers of sides so large that this becomes unwieldy, one uses a number, as in “regular

11

-gon” (as opposed to “

hendecagon

”), “regular

512

-gon” (as opposed to “pentacosiododecagon”) or “regular

n

-gon”. Below is a picture of some regular

n

-gons for

n

from

3

to

7

:

{xy}

0;¡3pc,0pc¿:¡0pc,3pc¿::

\ar

@-c+(0.866025,1.5);c+(1.73205,0.)

\ar

@-c+(1.73205,0.);c+(0.,0.)

\ar

@-c+(0.,0.);c+(0.866025,1.5)

triangle

{xy}

0;¡3pc,0pc¿:¡0pc,3pc¿::

\ar

@-c+(0,1.41421);c+(1.41421,1.41421)

\ar

@-c+(1.41421,1.41421);c+(1.41421,0.)

\ar

@-c+(1.41421,0.);c+(0.,0.)

\ar

@-c+(0.,0.);c+(0.,1.41421)
square

{xy}

0;¡3pc,0pc¿:¡0pc,3pc¿::

\ar

@-c+(0.951057,1.80902);c+(1.90211,1.11803)

\ar

@-c+(1.90211,1.11803);c+(1.53884,0.)

\ar

@-c+(1.53884,0.);c+(0.363271,0.)

\ar

@-c+(0.363271,0.);c+(0.,1.11803)

\ar

@-c+(0.,1.11803);c+(0.951057,1.80902)

pentagon

{xy}

0;¡3pc,0pc¿:¡0pc,3pc¿::

\ar

@-c+(0.5,1.73205);c+(1.5,1.73205)

\ar

@-c+(1.5,1.73205);c+(2.,0.866025)

\ar

@-c+(2.,0.866025);c+(1.5,0.)

\ar

@-c+(1.5,0.);c+(0.5,0.)

\ar

@-c+(0.5,0.);c+(0.,0.866025)

\ar

@-c+(0.,0.866025);c+(0.5,1.73205)

hexagon

{xy}

0;¡3pc,0pc¿:¡0pc,3pc¿::

\ar

@-c+(0.974928,1.90097);c+(1.75676,1.52446)

\ar

@-c+(1.75676,1.52446);c+(1.94986,0.678448)

\ar

@-c+(1.94986,0.678448);c+(1.40881,0)

\ar

@-c+(1.40881,0);c+(0.541044,0.)

\ar

@-c+(0.541044,0.);c+(0.,0.678448)

\ar

@-c+(0.,0.678448);c+(0.193096,1.52446)

\ar

@-c+(0.193096,1.52446);c+(0.974928,1.90097)

heptagon

If

n

is an

integer

with

n

≥

3

, then the

symmetry group

of a regular

n

-gon is the

n

\text

⁢

t

⁢

h

dihedral group

(written

D

n

). Unfortunately, since it has

2

⁢

n

elements, it is also sometimes called the dihedral group of order (

http://planetmath.org/OrderGroup

)

2

⁢

n

and written

D

2

⁢

n

.

In

spherical geometry

, every

biangle

is regular. Its symmetry group is the

Klein 4-group

.

The

center

of a regular polygon is the point that is equidistant from each of its vertices. (In spherical geometry, in to obtain uniqueness, we use the convention that the center is inside the polygon. Similarly, for biangles, we also use the convention that the center is equidistant from the

midpoints

of the sides.) An

apothem

of a regular polygon is a

line segment

such that all of the following occur:

•

one of its

endpoints

is the center of the regular polygon;

•

one of its endpoints is on one of the sides of the regular polygon;

•

it is

perpendicular

to the side of the regular polygon that it intersects.

For any regular polygon, all of its apothems are congruent, and the point at which an apothem and a side of the regular polygon intersect is the midpoint of that side.

Below is an example of a regular pentagon with all apothems drawn in blue.

From here on in the entry, only Euclidean geometry is considered.

Given a regular polygon with apothem of length

a

and

perimeter

(

http://planetmath.org/Perimeter2

)

P

, its area is

A

=

1

2

⁢

a

⁢

P

.

A proof of this statement is supplied in the entry

area of regular polygon

.

Any regular polygon can be

inscribed

into a circle and a circle can be inscribed within it. Note that both of these circles have the same center (

http://planetmath.org/Center8

) as the regular polygon. Below is a square with its inscribed circle drawn in green and its circumscribed circle drawn in cyan.

Given a regular polygon with

n

sides whose side has length

t

, the radius of the circumscribed circle is

R

=

t

2

⁢

sin

⁡

(

180

∘

n

)

and the radius of the inscribed circle is

r

=

2

⁢

t

⁢

tan

⁡

(

180

∘

n

)

.

The area of the regular polygon can also be calculated using the formula

A

=

n

⁢

t

2

4

⁢

tan

⁡

(

180

∘

n

)

.

In a more general setting, one might wish to allow figures which are not usually considered as polygons to be called “regular polygons”. In this sense, a regular

n

-gon is constructed from

n

line segments of the same length, each joined to the preceding one by the same angle on the same side, and the last joined to the first in the same way. It is not obvious, but any such figure can be made by choosing

n

points evenly spaced around a circle and joining them in some way. To get the ordinary convex

n

-gon, join each point to the next point. To obtain a more complicated

n

-gon, fix an integer

m

relatively prime to

n

with

1

<

m

<

n

-

1

and jump by

m

points each time. For any valid

k

, this procedure yields the same figure for

m

=

k

and

m

=

n

-

k

. Thus, there are

φ

⁢

(

n

)

2

of these generalized regular polygons (where

φ

denotes the

Euler totient function

) having

n

sides, including the ordinary convex

n

-gon. Since the line segments that form more complicated

n

-gons cross each other, there is no obvious definition of “area” to apply.

An example of a more complicated

n

-gon is a

pentagram

, which is obtained from the above procedure by letting

n

=

5

and either

m

=

2

or

m

=

3

. In the picture below, a pentagram is drawn in blue.

Title

regular polygon

Canonical name

RegularPolygon

Date of creation

2013-03-22 12:24:21

Last modified on

2013-03-22 12:24:21

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

48

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 51-00

Synonym

regular n-gon

Synonym

regular

Related topic

BasicPolygon

Related topic

Polyhedron

Related topic

RegularPolyhedron

Related topic

Pi

Related topic

CompassAndStraightedgeConstructionOfRegularPentagon

Related topic

RegularTriangle

Related topic

Square

Related topic

CriterionForConstructibilityOfRegularPolygon

Defines

apothem

Defines

center

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