Regular triacontagon
A regular triacontagon
Edge Edge (geometry) In geometry, an edge is a one-dimensional line segment joining two adjacent zero-dimensional vertices in a polygon. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects.... s and vertices Vertex (geometry) In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...	30
Schläfli symbol	{30}
Coxeter–Dynkin diagram
Symmetry group Symmetry group The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...	Dihedral (D30)
Internal angle Internal angle In geometry, an interior angle is an angle formed by two sides of a polygon that share an endpoint. For a simple, convex or concave polygon, this angle will be an angle on the 'inner side' of the polygon... (degree Degree (angle) A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians... s)	168°
Properties	convex Convex polygon In geometry, a polygon can be either convex or concave .- Convex polygons :A convex polygon is a simple polygon whose interior is a convex set... , cyclic, equilateral Equilateral In geometry, an equilateral polygon is a polygon which has all sides of the same length.For instance, an equilateral triangle is a triangle of equal edge lengths... , isogonal, isotoxal

In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, an triacontagon is a thirty-sided polygon

Polygon

In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...

. The sum of any triacontagon's interior angles is 5040 degrees.

One interior angle in a regular

Regular polygon

A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...

 triacontagon is 168° meaning that one exterior angle would be 12°

The regular triacontagon is a constructible polygon

Constructible polygon

In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not....

, by an edge-bisection

Bisection

In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a bisector. The most often considered types of bisectors are the segment bisector and the angle bisector In geometry, bisection is the division of something into two equal...

 of a regular pentadecagon

Pentadecagon

In geometry, a pentadecagon is any 15-sided, 15-angled, polygon.- Regular pentadecagon:A regular pentadecagon has interior angles of 156°, and with a side length a, has an area given by...

, and can be seen as a truncated

Truncation (geometry)

In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex.- Uniform truncation :...

 pentadecagon.

Area

The area

Area

Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

 of a regular triacontangon is (with t = edge length)

Petrie polygons

The regular triacontagon is the Petrie polygon

Petrie polygon

In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon such that every consecutive sides belong to one of the facets...

 for a number of higher dimensional polytopes with E₈ symmetry, shown in orthogonal projections in the E₈ Coxeter plane:

(421)

t1(421)

t2(421)

(241)

t1(241)

It is also the Petrie polygon for some higher dimensional polytopes with H₄ symmetry, shown in orthogonal projections in the H₄ Coxeter plane:

120-cell

Rectified 120-cell Rectified 120-cell In geometry, a rectified 120-cell is a uniform polychoron formed as the rectification of the regular 120-cell.There are four rectifications of the 120-cell, including the zeroth, the 120-cell itself...

Rectified 600-cell Rectified 600-cell In geometry, a rectified 600-cell is a uniform polychoron formed as the rectification of the regular 600-cell.There are four rectifications of the 600-cell, including the zeroth, the 600-cell itself...

600-cell