Cylinders and Prisms

Introduction: We examine cylinders and prisms, defining them, and measuring their surface areas and volumes.
The Lesson:

We may think of a cylinder as a tin can, with two bases that are circles. The surface wrapping around the circles can be “unwrapped” and shown as a rectangle. A diagram below shows a cylinder on the left. The rectangle at right is the side surface, unwrapped from the cylinder. We know that its length is the circumference of the circular base which is .

The surface area is the sum of the areas of the two circles that are the bases and the area of the rectangle. Since the area of a circle is and the area of a rectangle is lw, we can make a formula for the surface area of a cylinder: .

The volume of a cylinder is found using the concept of base x height. The area of the base is . The volume of the cylinder is .

A prism is a (3-dimensional) polyhedron with bases that are parallel, congruent polygons. The sides are parallelograms. We show three examples below. The important features are the areas of the bases and the height, which is the perpendicular distance between the bases.

At the left is a rectangular solid. This is discussed in (LINK myfilename Geom3D-rectsolids). The middle diagram is of a prism with bases that are hexagons. Notice that the slant height, s, is different than the height, h, since the prism is tilted. The prism at right has bases that are pentagons and is not tilted. We shall concentrate on prisms such as this which are called right prisms.

Volume and surface area can be found as follows:

• Volume = base (area) x height
• Surface Area = sum of areas of both bases and all sides

The area of the bases can be found if we know the dimensions and properties of the polygons which form the bases. The area of all the sides (called the lateral area) can be found if we can measure the base and height (called the slant height and labeled “s”) of each of the parallelograms which form the sides of the prism. Adding the areas of the parallelograms which form the sides can be summarized by multiplying the slant height and the perimeter of a base. We have:

• Surface Area = sum of areas of both bases and s x P where s is the slant height and P is the perimeter of a base.

If the bases are perpendicular to the parallelograms forming the sides (as in a right prism), the slant height s = h.

Let's Practice: We label the three prisms below A, B, and C. A is a rectangular solid. B is a non-right prism. C is a right prism.

1. Suppose prism A has dimensions 5 x 4 x 7 feet. What are the surface area and volume?

We solve this using standard formulas for a rectangular solid. (LINK myfilename Geom3D-rectsolids)

The volume is 5 x 4 x 7 = 140 cubic feet. Two sides are 5 x 4, two are 4 x 7 and two are 5 x 7. The areas of these sides are 20, 28, and 35.

Thus the total surface area is twice the sum of these or 166 square feet.

2. Suppose prism B has bases which are regular hexagons of side 2 meters, a height of 3 meters, and a slant height of 3.4 meters. What are the surface area and the volume?

The lateral area equals

s x P = 3.4 x 12 = 40.8 m².

The area of a regular hexagon (LINK myfilename GeomAreaPer-regpoly) (base) is

m².

This gives a total surface area of two bases plus the lateral area which equals

m².

We now know the volume = area of base x height equals

m³.

3. Suppose right prism C has bases which are regular pentagons of radius 25 centimeters. The height is 40 centimeters. What are the surface area and the volume?

For the area of a base (LINK myfilename GeomAreaPer-regpoly) we use

.

This gives us area of base as

cm².

We calculate volume = area of base x height as

1486 x 40 = 59440 cm³ = 0.05944 m³.

The lateral area is s x P where P is the perimeter of the base of the regular pentagon with radius 25 (LINK myfilename GeomAreaPer-regpoly) and s = h is the slant height.

A side can be found from

.

This gives us the length of a side of the base as

cm.

The lateral area equals

s x P = 40 x 29.39 = 1175.6 cm².

The total surface area is two bases plus the lateral area or

4147.6 cm2 = 0.41476 m².

4. A tin can has a radius of 2 inches and a height of 3 inches. What are the surface area and volumes?

The volume is

in³.

The surface area is

in².