Gúla

Gúla, in geometry, is a polyhedron consisting of a base polygon and triangular faces that meet at a single point called the apex. The base can be any polygon with at least three sides, and the apex is not in the plane of the base. The triangular faces connect each base vertex to the apex, forming a solid with as many lateral faces as the base has sides. The base determines the type of gúla, such as triangular, square, or pentagonal gúla.

A gúla is regular if its base is a regular polygon and the apex lies above the base center, so all lateral faces are congruent isosceles triangles. A right gúla is one whose apex projects perpendicularly onto the base center; a right regular gúla has its apex directly above the base center and all lateral faces identical.

Volume and surface area are standard measures for gúla. The volume equals one third of the base area times the height, V = (1/3) B h, where B is the area of the base and h is the perpendicular distance from the apex to the base. The surface area equals the base area plus the lateral area, SA = B + L, with the lateral area L = (1/2) P l for a regular gúla, where P is the base perimeter and l is the slant height. The slant height satisfies l^2 = h^2 + a^2, where a is the apothem of the base (the distance from the base center to a side).

Famous examples include the ancient Egyptian square-based pyramids. Gúla is a fundamental solid in classical and architectural geometry, illustrating how a base polygon gives rise to a symmetric, peaked form.