| Regular pentadecagon | |
|---|---|
 A regular pentadecagon | |
| Type | Regular polygon |
| Edges and vertices | 15 |
| Schläfli symbol | {15} |
| Coxeter–Dynkin diagrams |    |
| Symmetry group | Dihedral (D15), order 2×15 |
| Internal angle (degrees) | 156° |
| Properties | Convex, cyclic, equilateral, isogonal, isotoxal |
In geometry, a pentadecagon or pentakaidecagon or 15-gon is a fifteen-sided polygon.
A regular pentadecagon is represented by Schläfli symbol {15}.
A regular pentadecagon has interior angles of 156°, and with a side length a, has an area given by
A regular triangle, decagon, and pentadecagon can completely fill a plane vertex. However, due to the triangle's odd number of sides, the figures cannot alternate around the triangle, so the vertex cannot produce a semiregular tiling.
As 15 = 3 × 5, a product of distinct Fermat primes, a regular pentadecagon is constructible using compass and straightedge: The following constructions of regular pentadecagons with given circumcircle are similar to the illustration of the proposition XVI in Book IV of Euclid's Elements. [1]
Compare the construction according Euclid in this image: Pentadecagon
In the construction for given circumcircle: is a side of equilateral triangle and is a side of a regular pentagon. [2] The point divides the radius in golden ratio:
Compared with the first animation (with green lines) are in the following two images the two circular arcs (for angles 36° and 24°) rotated 90° counterclockwise shown. They do not use the segment , but rather they use segment as radius for the second circular arc (angle 36°).
A compass and straightedge construction for a given side length. The construction is nearly equal to that of the pentagon at a given side, then also the presentation is succeed by extension one side and it generates a segment, here which is divided according to the golden ratio:
Circumradius Side length Angle
The regular pentadecagon has Dih15 dihedral symmetry, order 30, represented by 15 lines of reflection. Dih15 has 3 dihedral subgroups: Dih5, Dih3, and Dih1. And four more cyclic symmetries: Z15, Z5, Z3, and Z1, with Zn representing π/n radian rotational symmetry.
On the pentadecagon, there are 8 distinct symmetries. John Conway labels these symmetries with a letter and order of the symmetry follows the letter. [3] He gives r30 for the full reflective symmetry, Dih15. He gives d (diagonal) with reflection lines through vertices, p with reflection lines through edges (perpendicular), and for the odd-sided pentadecagon i with mirror lines through both vertices and edges, and g for cyclic symmetry. a1 labels no symmetry.
These lower symmetries allows degrees of freedoms in defining irregular pentadecagons. Only the g15 subgroup has no degrees of freedom but can seen as directed edges.
There are three regular star polygons: {15/2}, {15/4}, {15/7}, constructed from the same 15 vertices of a regular pentadecagon, but connected by skipping every second, fourth, or seventh vertex respectively.
There are also three regular star figures: {15/3}, {15/5}, {15/6}, the first being a compound of three pentagons, the second a compound of five equilateral triangles, and the third a compound of three pentagrams.
The compound figure {15/3} can be loosely seen as the two-dimensional equivalent of the 3D compound of five tetrahedra.
| Picture |  {15/2}   |  {15/3} or 3{5} |  {15/4}  |  {15/5} or 5{3} |  {15/6} or 3{5/2} |  {15/7}  |
|---|---|---|---|---|---|---|
| Interior angle | 132° | 108° | 84° | 60° | 36° | 12° |
Deeper truncations of the regular pentadecagon and pentadecagrams can produce isogonal (vertex-transitive) intermediate star polygon forms with equal spaced vertices and two edge lengths. [4]
| Vertex-transitive truncations of the pentadecagon | ||||||||
|---|---|---|---|---|---|---|---|---|
| Quasiregular | Isogonal | Quasiregular | ||||||
 t{15/2}={30/2} |  |  |  |  |  |  |  |  t{15/13}={30/13} |
 t{15/7} = {30/7} |  |  |  |  |  |  |  | t{15/8}={30/8} |
 t{15/11}={30/22} |  |  |  |  |  |  |  | t{15/4}={30/4} |
The regular pentadecagon is the Petrie polygon for some higher-dimensional polytopes, projected in a skew orthogonal projection:
 14-simplex (14D) |
In geometry, a regular icosahedron is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces.
In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces.
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted .
In geometry, a hexagon is a six-sided polygon or 6-gon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.
In geometry, an octagon is an eight-sided polygon or 8-gon.
In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisectors of the triangle cross, as the point equidistant from the triangle's sides, as the junction point of the medial axis and innermost point of the grassfire transform of the triangle, and as the center point of the inscribed circle of the triangle.
In geometry, a decagon is a ten-sided polygon or 10-gon. The total sum of the interior angles of a simple decagon is 1440°.
In Euclidean geometry, a regular polygon is a polygon that is equiangular and equilateral. Regular polygons may be either convex or star. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon, if the edge length is fixed.
In geometry, an icosagon or 20-gon is a twenty-sided polygon. The sum of any icosagon's interior angles is 3240 degrees.
In geometry, a dodecagon or 12-gon is any twelve-sided polygon.
In geometry, a tridecagon or triskaidecagon or 13-gon is a thirteen-sided polygon.
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted ABCD.
In geometry, a triacontagon or 30-gon is a thirty-sided polygon. The sum of any triacontagon's interior angles is 5040 degrees.
In plane geometry, Morley's trisector theorem states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, called the first Morley triangle or simply the Morley triangle. The theorem was discovered in 1899 by Anglo-American mathematician Frank Morley. It has various generalizations; in particular, if all of the trisectors are intersected, one obtains four other equilateral triangles.
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
In mathematics, a hexadecagon is a sixteen-sided polygon.
In geometry, a pentagon is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
In geometry, an icositetragon or 24-gon is a twenty-four-sided polygon. The sum of any icositetragon's interior angles is 3960 degrees.
In geometry, a planigon is a convex polygon that can fill the plane with only copies of itself. In the Euclidean plane there are 3 regular planigons; equilateral triangle, squares, and regular hexagons; and 8 semiregular planigons; and 4 demiregular planigons which can tile the plane only with other planigons.
