Big Picture

Prisms and cylinders are among the simplest 3-dimensional objects. They have two parallel bases of equal size. We can find the volume and surface area of these objects by using formulas.

Key Terms

Prisms

A prism has a pair of parallel bases and rectangular lateral (non-base) faces. Prisms are named by their bases. If the lateral faces are perpendicular to the bases, the prism is a right prism. If the faces lean to one side, the prism is oblique.

Surface of a Prism

Theorem: The surface area of a prism is the sum of the area of the bases and the area of each lateral face.

• surface area = SA = lateral area + 2 • area of base

The lateral area is the sum of the area of all the lateral faces. Another way to calculate the lateral area for a right prism is:

• lateral area = perimeter • height

Another way to write the surface area for a right prism is:

• SA = perimeter • height + 2 • area of base

Volume of a Prism

Theorem: The volume of a prism is V = Bh, where B is the area of the base and h is the prism’s height.

For a rectangular prism, the base is a rectangle, so the volume formula can be rewritten as: V = lwh.

For oblique prisms, Cavalieri’s Principle holds, so the volumes of oblique prisms and right prisms have the same formula. The height for oblique prisms is the altitude outside the prism.

Cylinders

A cylinder is a 3-dimensional figure with a pair of parallel and congruent circular ends, or bases.

Cylinders (cont.)

Surface Area of a Right Cylinder

We can deconstruct a cylinder into a net. The sum of the areas of all the components, the two bases and the lateral side, will give us the total surface area of the cylinder.

The net for a right cylinder is a rectangle and two circular bases. The surface area can be found with the following equation:

     SA = (area of two bases) + (area of lateral side)

This can also be written as:

     \(SA = 2 • (πr²)+2πr • h\)

Theorem: The surface area of a right cylinder with radius r and height h is \(SA = 2 \pi r^2 + 2 \pi rh\).

Volume of a Cylinder

Theorem: The volume of a cylinder is V = bh, where b is the area of the base and a is the prism’s height.

The base of a cylinder is a circle, so the volume of a cylinder can be written as: \(V = \pi r^2 h\).

For oblique cylinders, Cavalieri’s Principle holds, so the volumes of oblique cylinders and right cylinders have the same formula. The height for oblique cylinders is the altitude outside the prism.

Notes