Pyramid

A pyramid is a geometric solid of the shape made famous by the royal tombs of ancient Egypt. It is a solid whose base is a polygon and whose lateral faces are triangles with a common vertex (the vertex of the pyramid). In the case of the Egyptian pyramid of Cheops, the base is an almost perfect square 755 ft (230 m) on an edge, and the faces of triangles that are approximately equilateral.

Pyramids at Giza, Egypt. Photograph by Dilip Mehia/Contact Giza. Stock Market. Reproduced by permission.

The base of a pyramid can be any polygon of three or more edges, and pyramids are named according to the number of edges in the base. When the base is a triangle, the pyramid is a triangular pyramid. It is also known as a tetrahedron since, including the base, it has four faces. When these faces are equilateral triangles, it is a square pyramid, having a square as its base.

The pyramids most commonly encountered are "regular" pyramids. These have a regular polygon for a base and isosceles triangles for lateral faces. Not all pyramids are regular, however.

The height of a pyramid can be measured in two ways, from the vertex along a line perpendicular to the base and from the vertex along a line perpendicular to one of the edges of the base. This latter measure is called the slant height. Unless the lateral faces are congruent triangles, however, the slant height can vary from face to face and will have little meaning for the pyramid as a whole. Unless the word slant is included, the term height (or altitude) refers to the height.

If in addition to being congruent, the lateral faces are isosceles, the pyramid will be regular. In a regular pyramid, right triangles are to be found in abundance. Suppose we have a regular pyramid whose altitude is VC and slant height VD. Here the triangles VCD, VDE, VCE, and CDE are all right triangles. If in any of these triangle one knows two of the sides, one can use the Pythagorean theorem to figure out the third. This, in turn, can be used in other triangles to figure out still other unknown sides. For example, if a regular square pyramid has a slant height of two units and a base of two units on an edge, the lateral edges have to be √5 units and the altitude √3 units.

There are formulas for computing the lateral area and the total area of certain special pyramids, but in most instances it is easier to compute the areas of the various faces and add them up.

Volume is another matter. Figuring volume without a formula can be very difficult. Fortunately there is a rather remarkable formula dating back at least 2,300 years.

In Proposition 7 of Book XII of his Elements, Euclid showed that "Any prism which has a triangular base is divided into three pyramids equal to one another which have triangular bases." This means that each of the three pyramids into which the prism has been divided has one third the prism's volume. Since the volume of the prism is the area, B, of its base times its altitude, h: the volume of the pyramid is one third that, or Bh/3.

Pyramids whose bases are polygons of more than three sides can be divided into triangular pyramids and Euclid's formula applied to each. Then if B is the sum of the areas of the triangles into which the polygon has been divided, the total volume of the pyramid will again be Bh/3.

If one slices the top off a pyramid, one truncates it. If the slice is parallel to the base, the truncated pyramid is called a frustum. The volume of a frustum is given by the curious formula (B + B' + √BB )h/3, where B and B' are the areas of the upper and lower bases, and h is the perpendicular distance between them.