Volume of a Cylinder

We know that the volume of a three-dimensional solid is the amount of space it occupies. We also know that a cylinder is a solid composed of two congruent circles in parallel planes, their interiors, and all of the line segments parallel to the segment containing the centers of both circles with endpoints on the circular regions. Have you ever considered how to find the volume of a cylinder?

The formula for the volume of a cylinder is $V=Bh$ where V represents the volume, B is the area of the circle at its base, and h is the height of the cylinder. If you're given measurements for the area of the base and the height, you can use this formula directly. However, you usually won't be given the area of the base. Instead, you'll be given something like this:

We don't know what the area of the base is, but we do have the radius of the circle at the base and the height of the cylinder. This article will show you another formula you can use to find the volume of a cylinder with only that information. Yes, it involves one of our favorite numbers, pi.

Finding the volume of a cylinder

The formula we'll be using to find the volume of a cylinder is as follows:

$V=\pi r^{2}h$

In this formula, V represents the volume of the cylinder, π is pi, r is the radius of the circle at the base of the cylinder, and h is the height. You may recall that the formula for finding the area of a circle is $A=\pi r^2$ , so all this formula is doing is giving us the area of the circle and multiplying it by the height to give us the volume in a single equation. Subbing in the numbers from the diagram above, we get:

$V=\pi {\left(8\right)}^{2}15$

$V=\pi \left(64\right)15$

$V\approx 3016{\mathrm{cm}}^3$

Remember that volume is always expressed in cubic units such as ${\mathrm{in}}^3,m^3,{\mathrm{ft}}^3$ , and so on. Otherwise, finding the volume of a cylinder is as simple as memorizing the formula above and plugging in the numbers where they are supposed to go.

You may also be given the diameter of the circle at the base of a cylinder instead of its radius. A circle's radius is always half of its diameter, so cutting the provided diameter in half will give you the radius you need to use the formula. For example, let's say you're working with a cylinder that has a height of 4 inches and a diameter of 10 inches. The radius would be 5 inches (half of the diameter), giving us all of the numbers we need to use the formula:

$V=\pi {\left(5\right)}^2\left(4\right)$

$V=\pi \left(25\right)\left(4\right)$

$V\approx 314{\mathrm{in}}^3$

That wasn't too bad, was it? Some problems might treat pi differently as well. For instance, you may be asked to use 3.14 as the value of π, use the actual value of pi (in which case you will probably be allowed to use a calculator), or express your answer in terms of π (treating it like the "x" in 3x). Just be sure to read the question carefully so you don't miss any important information.

Practice questions on finding the volume of a cylinder

a. What is the volume of a cylinder with a radius of 10 meters and a height of 17 meters? Use 3.14 as the value of pi.

$V=\pi r^{2}h$

$V=3.14\times {10}^2\times 17$

$V=3.14\times 100\times 17$

$V=5338m^3$

b. What is the volume of a cylinder with a diameter of 40 cm and a height of 10 cm? Use 3.14 as the value of pi.

First, we need to find the radius, which is half of the diameter.

$\text{Radius}=\frac{\text{Diameter}}{2}=\frac{40\mathrm{\hspace{0.17em}cm}}{2}=20\mathrm{\hspace{0.17em}cm}$

$V=\pi r^{2}h$

$V=3.14\times {20}^2\times 10$

$V=3.14\times 400\times 10$

$V=12560{\mathrm{cm}}^3$

c. What is the volume of a cylinder with a height of seven inches and a base measuring 14 in²?

In this problem, we're given the area of the base (A) instead of the radius. We can use the formula $V=A\times h$ .

$V=A\times h$

$V=14{\mathrm{in}}^2\times 7\mathrm{in}$

$V=98{\mathrm{in}}^3$

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