In certain prisms, the lateral faces are each perpendicular to the plane of the base (or bases if there is more than one). These are known as a group as right prisms.

The lateral area of a right prism is the sum of the areas of all the lateral faces.

*Theorem 87:* The lateral area, *LA*, of a right prism of altitude *h* and perimeter *p* is given by the following equation.

**Example 1:** Find the lateral area of the right hexagonal prism, shown in Figure 1.

**Figure 1 **A right hexagonal prism.

The total area of a right prism is the sum of the lateral area and the areas of the two bases. Because the bases are congruent, their areas are equal.

*Theorem 88:* The total area, *TA*, of a right prism with lateral area *LA* and a base area *B* is given by the following equation.

**Example 2:** Find the total area of the triangular prism, shown in Figure 2.

**Figure 2 **A (right) triangular prism.

The base of this prism is a right triangle with legs of 3 ft and 4 ft (Figure 3).

**Figure 3 **The base of the triangular prism from Figure 2.

The perimeter of the base is (3 + 4 + 5) ft, or 12 ft.

Because the triangle is a right triangle, its legs can be used as base and height of the triangle.

The altitude of the prism is given as 2 ft. Therefore,

Lateral area and total area are measurements of the surface of a solid. The interior space of a solid can also be measured.

A **cube** is a square right prism whose lateral edges are the same length as a side of the base; see Figure 4.

**Figure 4 **A cube.

The **volume** of a solid is the number of cubes with unit edge necessary to entirely fill the interior of the solid. In Figure 5, the right rectangular prism measures 3 inches by 4 inches by 5 inches.

**Figure 5 **Volume of a right rectangular prism.

This prism can be filled with cubes 1 inch on each side, which is called a **cubic inch**. The top layer has 12 such cubes. Because the prism has 5 such layers, it takes 60 of these cubes to fill this solid. Thus, the volume of this prism is 60 cubic inches.

*Theorem 89:* The volume, *V*, of a right prism with a base area *B* and an altitude *h* is given by the following equation.

**Example 3:** Figure 6 is an isosceles trapezoidal right prism. Find (a) *LA* (b) *TA* and (c) *V*.

**Figure 6 **An isosceles trapezoidal right prism.

*LA* _{right prism} = ( *p*)( *h*) units^{2}

( Note: The *h* refers to the altitude of the prism, not the height of the trapezoid.)

*V* _{right prism} = ( *B*)( *h*) units^{3}

( Note: The *h* refers to the altitude of the prism, not the height of the trapezoid.)