Calculate the Volume of a Cylinder

By Mark Ryan

To calculate the volume of a cylinder, you need to know its height and the area of its base. Because a cylinder is a flat-top figure (a solid with two congruent, parallel bases), the base can be either the top or bottom.

If you know a cylinder’s height and lateral area, but not its radius, you can use the formula for surface area to find the radius, and then calculate the volume from there.

The lateral area of a cylinder is basically one rectangle rolled into a tube shape. Think of the lateral area of a cylinder as one rectangular paper towel that rolls exactly once around a paper towel roll. The base of this rectangle (you know, the part of the towel that wraps around the bottom of the roll) is the same as the circumference of the cylinder’s base. And the height of the paper towel is the same as the height of the cylinder.

geometry-cylinder

You can find the volume of a cylinder using the following formula:

geometry-cylinder-formula

Now for a cylinder problem:

geometry-cylinder-problem

Here’s a diagram to help you.

Geometry-volume-diagram

To use the volume formula, you need the cylinder’s height (which you know) and the area of its base. To get the area of the base, you need its radius. And to get the radius, you can use the surface area formula and solve for r:

geometry-radius

Remember that this “rectangle” is rolled around the cylinder and that the “rectangle’s” base is the circumference of the cylinder’s circular base. You fill in the equation as follows:

geometry-circumference

Now set the equation equal to zero and factor:

geometry-equal-zero

The radius can’t be negative, so it’s 5. Now you can finish with the volume formula:

geometry-volume-finish

That does it.