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Prism

A prism is a three-dimensional figure or polyhedron having two faces (called the bases of the prism) which are congruent polygons, and the remaining faces parallelograms .

Prisms

Prisms

If the remaining faces are rectangles (as in the figure 2 nd figure above), then the prism is called a right prism . School level math books usually only talk about right prisms, and sometimes when they say prism they mean right prism.

A triangular prism is a prism with a triangular base, like the figure on the right. A rectangular prism is a prism with a rectangular base. (The cube is a special case.) Similarly, a prism with a 5 -sided base is a pentagonal prism , a prism with a 6 -sided base is a hexagonal prism , etc.

Volume of a Prism

The volume V of a right prism is given by the formula

V = B h ,

where B is the area of the base and h is the height.

Example:

Find the volume of the prism shown.

Right Triangular Prism

The height of the prism is 10 cm. The base is a right triangle with legs of length 5 cm and 12 cm, so

B = 1 2 ( 5 ) ( 12 )           = 1 2 ( 60 )           = 30  cm 2

Therefore, the volume is

V = B h           = 30 ( 10 )           = 300  cm 3

Note that for a rectangular prism, the volume formula becomes simply

V = l w h ,

where l is the length of the base, w is the width of the base, and h is the height of the prism.

Surface Area of a Prism

The surface area S of a prism is given by the formula

S = 2 B + P h ,

where B is the area of the base of the prism, P is the perimeter of the base, and h is the height.

In the example above, by the Pythagorean theorem , the hypotenuse of the triangular base has length

5 2 + 12 2 = 25 + 144                                           = 169                                           = 13  cm

So the perimeter of the base is 5 + 12 + 13 = 30 cm. We already know the base area is 30 cm 2 . So,

S = 2 B + P h         = 2 ( 30 ) + ( 30 ) ( 10 )         = 60 + 300         = 360  cm 2

You may sometimes be called upon to find the lateral surface area of a prism. This just means the surface area of the faces not including the bases, or L = P h .