The base of a regular pyramid is a regular polygon and its lateral edges are equal in length. (Therefore the lateral faces are are congruent isosceles triangles.) In regular pyramids the base point of the height coincides with the center of the base.
The base of a regular square pyramid is a regular quadrilateral, i.e. a square. Its base edges (a) are equal in length and so are the lateral edges (b). Therefore, the lateral faces are isosceles triangles. In regular pyramids the base point of the height coincides with the center of the square base (O).
The lateral faces of the pyramid form the lateral surface, which, in the case of a regular square pyramid, comprises four congruent isosceles triangles. The area of such a triangle is half the product of its base length and its height. (In other words, it is half the product of the pyramid’s base edge (a) and the slant height (l).) The area of the base is the square (a²) of the length of base edges (a). The surface area of the pyramid is the sum of the base area and the lateral surface area (the sum of the areas of the lateral faces).
To calculate the volume of the pyramid, we can start with the volume of a prism with the same base and height as those of the pyramid. The volume of the prism is the product of its base area (A) and its height (h). The volume of the pyramid is one third the volume of the prism. In other words, the volume of the pyramid is one third the product of its base area and its height.