The number of vertices is 720 divided by the vertex angle defect.

The vertex angle of elytron is extended in a triangular spine.

Each edge is labeled by a rational value corresponding to the reflection order, being π/vertex angle.

The polar sine of the vertex angle is:

The value n/d means the vertex angle is d/n of the half-circle.

Robert Neale developed a system to model equilateral polyhedra based on a module with variable vertex angles.

In geometry, a vertex angle is the angle associated with a vertex of a polygon.

In Euclidean geometry, an equiangular polygon is a polygon whose vertex angles are equal.

This is due to the impossibility of tiling a plane with regular pentagons, simply because the pentagon vertex angle does not divide .

The vertex angle (denoted by α or A) in a polygon is measured by the interior side of the vertex.

In this section, the vertex angles are labeled A, B, C and all coordinates are trilinear coordinates:

In geometry, the polar sine generalizes the sine function of angle to the vertex angle of a polytope.

In fiber optics, the radiation angle is half the vertex angle of the cone of light emitted at the exit face of an optical fiber.

Moreover the corresponding triangle center coincides with the obtuse angled vertex whenever any vertex angle exceeds 2π/3, and with the 1st isogonic center otherwise.

The sum of the vertex angles of spherical triangles is always larger than the sum of the angles of plane triangles, which is exactly 180 .

This is readily seen to be a triangle center function and (provided the triangle is scalene) the corresponding triangle center is the excenter opposite to the largest vertex angle.

A rhombus has an axis of symmetry through each pair of opposite vertex angles, while a rectangle has an axis of symmetry through each pair of opposite sides.

If a, b, c, d are the length of the sides AB, BC, CD, DA respectively in a bicentric quadrilateral ABCD, then its vertex angles can be calculated with the tangent function:

These are the internal angle bisectors at two opposite vertex angles, the external angle bisectors at the other two vertex angles, and the external angle bisectors at the angles formed where the extensions of opposite sides intersect.