# Prism

In Euclidean geometry, a prism is a three dimensional figure, or solid, having five or more faces, each of which is a polygon. Polygons, in turn, consist of any number of straight line segments, arranged to form a flat, closed, two-dimensional figure. Thus, triangles, rectangles, pentagons, hexagons, and so on are all polygons. In addition, a prism has at least two congruent (same size and shape) faces that are parallel to one another. These parallel faces are called bases of the prism, and are often associated with its top and bottom. An interesting property of prisms is that every cross section, taken parallel to a base, is also congruent to the base. The remaining faces of a prism, called lateral faces, meet in line segments called lateral edges. Every prism has as many lateral faces, and lateral edges, as its base has sides. Thus, a prism with an octagonal (eight sided) base has eight lateral faces, and eight lateral edges. Each lateral face meets two other lateral faces, as well as the two bases. As a consequence, each lateral face is a four sided polygon. It can also be shown that, because the bases of a prism are congruent and parallel, each lateral edge of a prism is parallel to every other lateral edge, and that all lateral edges are the same length. As a result, each lateral face of a prism is a parallelogram (a four-sided figure with opposite sides parallel).

There are three important special cases of the prism, they are the regular prism, the right prism, and the parallelepiped. First, a regular prism is a prism with regular polygon bases. A regular polygon is one that has all sides equal in length and all angles equal in measure. For instance, a square is a regular rectangle, an equilateral triangle is a regular triangle, and a stop sign is a regular octagon. Second, a right prism is one whose lateral faces and lateral edges are perpendicular (at right, or 90° angles) to it bases. The lateral faces of a right prism are all rectangles, and the height of a right prism is equal to the length of its lateral edge. The third important special case is the parallelepiped. What makes the parallelepiped special is that, just as its lateral sides are parallelograms, so are its bases. Thus, every face of a parallelepiped has four sides. A special case of the parallelepiped is the rectangular parallelepiped, which has rectangular bases (that is, parallelograms with 90° interior angles), and is sometimes called a rectangular solid. Combining terms, of course, leads to even more restricted special cases, for instance, a right, regular prism. A right, regular prism is one with regular polygon bases, and perpendicular, rectangular, lateral sides, such as a prism with equilateral triangles for bases and three rectangular lateral faces. Another special type of prism is the right, regular parallelepiped. Its bases are regular parallelograms. Thus, they have equal length sides and equal angles. For this to be true, the bases must be squares. Because it is a right prism, the lateral faces are rectangles. Thus, a cube is a special case of a right, regular, parallelepiped (one with square lateral faces), which is a special case of a right, regular prism, which is a special case of a regular prism, which is a special case of a prism.

The surface area and volume of a prism are two important quantities. The surface area of a prism is equal to the sum of the areas of the two bases and the areas of the lateral sides. Various formulas for calculating the surface area exist, the simplest being associated with the right, regular prisms. The volume of a prism is the product of the area of one base times the height of the prism, where the height is the perpendicular distance between bases.

## Resources

### Books

Smith, Stanley A., Charles W. Nelson, Roberta K. Koss, Mervin L. Keedy, and Marvin L Bittinger. Addison Wesley Informal Geometry. Reading MA: Addison Wesley, 1992.

Welchons, A.M., W.R. Krickenberger, and Helen R. Pearson. Plane Geometry. Boston, MA: Ginn and Company, 1965.

## KEY TERMS

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Base

—The base of a prism is one of two congruent, parallel sides, often used to indicate the top and bottom of the prism.

Edge

—An edge of a prism is the line formed by intersecting faces of the prism.

Lateral face

—The lateral faces of a prism are those faces other than the bases.

Parallelepiped

—A parallelepiped is a prism whose bases are parallelograms.

Regular prism

—A regular prism is one with regular polygons as bases.

Right prism

—A right prism is one with lateral sides perpendicular to its bases.

Surface area

—The surface area of a prism is equal to the number of unit squares it takes to tile the entire surface of the prism.

Volume

—The amount of space that a material body occupies.