In
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a tridecagon or triskaidecagon or 13-gon is a thirteen-sided
polygon
In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain.
The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
.
Regular tridecagon
A ''
regular
Regular may refer to:
Arts, entertainment, and media Music
* "Regular" (Badfinger song)
* Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings
Other uses
* Regular character, ...
tridecagon'' is represented by
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...
.
The measure of each internal angle of a
regular
Regular may refer to:
Arts, entertainment, and media Music
* "Regular" (Badfinger song)
* Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings
Other uses
* Regular character, ...
tridecagon is approximately 152.308
degrees, and the area with side length ''a'' is given by
:
Construction
As 13 is a
Pierpont prime
In number theory, a Pierpont prime is a prime number of the form
2^u\cdot 3^v + 1\,
for some nonnegative integers and . That is, they are the prime numbers for which is 3-smooth. They are named after the mathematician James Pierpont, who us ...
but not a
Fermat prime
In mathematics, a Fermat number, named after Pierre de Fermat (1601–1665), the first known to have studied them, is a positive integer of the form:F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: 3, 5, ...
, the regular tridecagon cannot be
constructed using a
compass and straightedge
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an Idealiz ...
. However, it is constructible using
neusis
In geometry, the neusis (; ; plural: ) is a geometric construction method that was used in antiquity by Greek mathematics, Greek mathematicians.
Geometric construction
The neusis construction consists of fitting a line element of given length ...
, or an angle trisector.
The following is an animation from a ''neusis construction'' of a regular tridecagon with radius of circumcircle
according to
Andrew M. Gleason
Andrew Mattei Gleason (19212008) was an American mathematician who made fundamental contributions to widely varied areas of mathematics, including the solution of Hilbert's fifth problem, and was a leader in reform and innovation in teaching a ...
,
based on the
angle trisection
Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and ...
by means of the
Tomahawk
A tomahawk is a type of single-handed axe used by the many Native Americans in the United States, Indian peoples and nations of North America, traditionally resembles a hatchet with a straight shaft.
Etymology
The name comes from Powhatan langu ...
(light blue).

An approximate construction of a regular tridecagon using
straightedge
A straightedge or straight edge is a tool used for drawing straight lines, or checking their straightness. If it has equally spaced markings along its length, it is usually called a ruler.
Straightedges are used in the automotive service and ma ...
and
compass
A compass is a device that shows the cardinal directions used for navigation and geographic orientation. It commonly consists of a magnetized needle or other element, such as a compass card or compass rose, which can pivot to align itself with No ...
is shown here.
Another possible animation of an approximate construction, also possible with using straightedge and compass.
Based on the unit circle r = 1
nit of length
NiT, NIT, or Nits may refer to:
Education
* Namgyal Institute of Tibetology, Sikkim, India
* Narula Institute of Technology, West Bengal, India
* National Institutes of Technology, India
* Naval Institute of Technology, Biliran, Philippines
* Nip ...
/h2>
* Constructed side length in
GeoGebra
GeoGebra (a portmanteau of ''geometry'' and ''algebra'') is an interactive geometry, algebra, statistics and calculus application, intended for learning and teaching mathematics and science from primary school to university level. GeoGebra is a ...