# Prisms and Cylinders

## 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

Prism: A polyhedron with two parallel, congruent bases.

Cylinder: A solid with congruent circular bases that are in parallel planes with the space between the circles is enclosed.

## 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.

# Prisms and Cylinders Cont.

## 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.