We are asked to describe how the volume of a cylinder and the volume of a rectangular prism are related. Let's remember how we calculate each volume. We will start with the formula for the volume of a cylinder.

Volume of a Cylinder

To calculate the volume of a cylinder, we multiply the area of the base by the height.

Note that if the radius of the circular base of a cylinder is $r,$ then the area of the base $B$ is $π r^{2} .$

Let's substitute this area into the formula for the volume.

V = B h \Rightarrow V = (π r^{2}) h

The result is a formula for the volume of a cylinder with a radius

r

and height

h .

Let's now remember the formula for the volume of a rectangular prism.

Volume of a Prism

Same as before, we calculate the volume by multiplying the area of the base by the height.

The only difference is that this time the base is a rectangle, not a circle. If $ℓ$ and $w$ are the length and width of the base, the area of the base is $ℓ \cdot w .$

Let's then substitute this expression for

B

in the formula.

V = B h \Rightarrow V = (ℓ w) h

We can use this formula to find the volume of a prism with a rectangular base.

Comparison

See that the rules to calculating the volumes are the same. In both cases we multiply the area of the base

B

by the height of the figure

h .

V_{cylinder} = B h vs V_{prism} = B h

The only difference are the areas of the bases. The base of a cylinder is a circle and the base of a rectangular prism is a rectangle. As a result, the final formulas are different.

V_{cylinder} = π r^{2} h vs V_{prism} = ℓ w h

Exercise