vertexangle

Vertex angle is the term used in geometry to describe the angle located at a vertex of a geometric figure. The vertex is the point where two or more edges, lines, or sides meet, and the vertex angle is the measure of the angle formed by those meeting elements. In two‑dimensional geometry the most common reference is the apex angle of an isosceles triangle. For an isosceles triangle the two congruent sides converge at the vertex, and the vertex angle is the angle between them; the remaining two angles, called base angles, are equal. In ad hoc contexts the vertex angle may simply refer to any angle at a vertex of a polygon or quadrilateral, often measured in degrees or radians.

In three‑dimensional geometry vertex angles appear at the apexes of pyramids, cones, and other solids. For a pyramid the vertex angle is typically measured between two slant edges that meet at the apex. For a cone the vertex angle is the angle between two generators that emanate from the apex and are separated by a given meridian; this angle determines the steepness of the cone. In polyhedra the internal dihedral angle between two faces meeting at an edge is sometimes called an edge angle, while the angle at a face’s vertex is the face vertex angle.

Vertex angles have practical importance in engineering, architecture, and computer graphics where the shape and stability of structures depend on the geometry of vertex angles. Calculations of vertex angles involve standard trigonometric relations such as the law of cosines and basic angle addition rules. Understanding and controlling vertex angles is essential for designing fair surfaces, symmetrical forms, and optimizing geometric configurations.