Volume & Surface Area of a Torus

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  • 0:00 Torus
  • 0:57 Volume & Surface Area…
  • 2:28 Example
  • 5:05 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

You've heard of squares, circles and triangles, but what is a torus? In this lesson, we will learn what a torus is in mathematics, and then we will see how to calculate its surface area and volume. We will also look at a real world example involving these processes.


Who likes doughnuts? I do, I do! They're just so yummy! Well, what if you were at a bakery, and the person at the counter asked you if you would like a torus for breakfast? Wait a minute, a what? I don't know about you, but that sounds kind of weird! No thanks! However, by turning down this offer, we may actually be missing out on a delicious doughnut for breakfast!

In mathematics, a torus is the name we use to describe the 3-dimensional shape of a doughnut. Of course, there are other shapes in the world around us that have the shape of a doughnut, not just a doughnut, so we would describe these things as toruses as well. For instance, the person at the counter that offered us a torus for breakfast could also be asking if we want a bagel for breakfast. Point being, torus is just the name that we use to describe objects that are doughnut shaped, not necessarily an actual doughnut.

Volume & Surface Area of a Torus

Before we get to the volume and surface area of a torus, let's first review what volume and surface area are. The volume of a 3-dimensional object refers to how much space the object takes up. The surface area of a three-dimensional object refers to the total area of the surface of the object.

To calculate the volume and surface area of a torus, we first need to know the inner and outer radius of the torus. The inner radius of a torus is the radius of the inner hole of the torus, and the outer radius of a torus is the radius of the entire object.

Now that we know what the inner and outer radius of a torus is, let's take a look at the formulas we can use to calculate the volume and surface area of a torus. To do this, let's let R be the outer radius of a torus and r be the inner radius of a torus.

To find the volume, we use the following formula:

Volume = (1/4)(π 2)(R 2 - r 2)(R - r)

To find the surface area, we use the following formula:

Surface Area = (R 2 - r 2 2

The formulas are a bit involved, but really, all we have to do to find the volume and surface area of a torus is measure its inner radius and its outer radius, and then plug those values into the formulas accordingly and simplify. That's not so bad, is it? Actually, let's go ahead and try it with an example.


Suppose you're going to head out on a tubing trip with some friends. You have to bring your own tube, and you found an old black tube that you can use in the garage. The tube has to meet specific criteria for the tubing company to allow it on the river. One of those criteria is that its volume must be no larger than 11,000 cubic inches, so the first thing you want to do is make sure that your tube fits this criteria so you can use it.

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