Regular Pyramids

A regular pyramid is a pyramid whose base is a regular polygon and whose lateral edges are all equal in length. A pyramid is named by its base. Figure 1 shows some examples of regular pyramids.

Figure 1 Some different types of regular pyramids.

The lateral faces of a regular pyramid are congruent isosceles triangles. The altitude of any of these triangles is the slant height of the regular pyramid. Figure 2 is a square pyramid.

Figure 2 A square pyramid.

Pyramids also have a lateral area, total area, and volume.

Theorem 93: The lateral area, LA, of a regular pyramid with slant height l and base perimeter p is given by the following equation.

Example 1: Find the lateral area of the square pyramid, shown in Figure 3 .

Figure 3 Finding the lateral area, total area, and volume of a square pyramid.

Because a pyramid has only one base, its total area is the sum of the lateral area and the area of its base.

Theorem 94: The total area, TA, of a regular pyramid with lateral area LA and base area B is given by the following equation.

Example 2: Find the total area of the regular pyramid shown in Figure 3 .

The base of the regular pyramid is a square. A_square = (side)². Therefore, B = 16² in², or B = 256 in².

From the previous example,

Theorem 95: The volume, V, of a regular pyramid with base area B and altitude h is given by the following equation.

Example 3: Find the volume of the regular pyramid shown in Figure 3 .

From the previous example, B = 256 in². The figure indicates that h = 6 in.

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