Word Lesson: Volume and Surface Area of Prisms

By M Ransom

We will work with right prisms, in which the slant height is the same as the height. In order to solve problems which require application of the volume and surface area for prisms, it is necessary to

know how to use basic formulas for volume (V) and surface area (SA) of a right prism:

V = area of base x height

SA = sum of both bases and the lateral area

know how to find the area of a base:

regular polygon: base area =

regular hexagon: base area =

know how to find the length of a side of a regular polygon using
solve basic linear and quadratic equations.

A typical problem involving the volume or surface area of a prism gives us one or more of the volume, lateral area, area of a base, height and/or radius of the prism. We will be required to calculate some of these quantities given information about the others.

Suppose the height of a right rectangular prism (rectangular solid) is 30 cm and the volume is 480 cm³. If the base is a square, find the surface area of this prism. A diagram is shown below.

To get started, we sketch a diagram and label all of the given information to determine the appropriate formula(s) we will be able to use.

Since we know the volume is 480 cm³, we will start with the formula for volume.

V = 480 cm³
V = lwh

The length l and width w are both the same in a square base allowing us to use the variable x for both l and w. Because h is given as 30 cm we can write:

V = (x)(x)(30)
V = 30x²
30x² = 480
x² = 16
x = 4 cm

We will now use x = 4 cm and h = 30 cm to calculate the area of all six rectangular sides of this prism. Remember that the 2 bases are equal as are the four sides making up the lateral area.

There are two square bases. Since the length of a side x = 4, there is a total area of

2x² = 2(4)² = 32 cm²
The four sides are rectangles with dimensions of width 4 cm and height 30 cm for a total area of

4(4)(30) = 480 cm²
The total surface area is

32 cm²+ 480 cm² = 512 cm²

A right prism with regular hexagonal bases has a height of 20 inches and a radius of 5 inches. What are the surface area and volume?

A right prism has a regular pentagon of radius 7 for a base and a height of 10. What are the volume and surface area of this prism?

A right prism has a regular hexagon of radius 12 as a base. If the height is 10, what are the volume and surface area?

V = and SA =

V = and SA =

V = and SA =

A right prism has a height of 14 and a regular octagonal base of radius 9. What are the volume and surface area of this prism?

V = 357 and S = 822.4
V = 3207.4 and S = 1229.6
V = 275.7 and S = 1534.2

This type of problem involves the use of several formulas. If the prism is neither rectangular nor hexagonal, we must use formulas involving sine and cosine to get the area of the base and the length of one side of the base. If rectangular, we can use length x width for areas of rectangular sides and bases. If hexagonal, we have that the radius and length of a side of the base are the same and the area of the base is given by .

In any case, the base area is used to get the volume when we multiply it by the prism's height. The side of the base is used to get the perimeter which is then used to get lateral area, LA = h x P, where h is the height of the prism.

Since sine and cosine are sometimes used, it is important to have your calculator set for either radians or degrees depending on your given information. Usually calculations can left in the degree MODE since our basic formulas involve 180/n where 180 is in degrees.

It is difficult to check these answers for reasonableness and therefore very important to double check all arithmetic carefully.