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.

## Torus

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)π 2The 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.

## 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.The first thing you notice is that the tube has the shape of a torus. Great! You know how to calculate the volume of this! You take measurements of the inner and outer radius to find that the inner radius is *r* = 9 inches, and the outer radius is *R* = 20 inches.

Okay, let’s get to work by plugging these values into our volume formula to see if you’ll be able to use this tube. We work out our calculations, simplifying as we go:

We see that the volume of the tube is approximately 8,658.11 cubic inches, which is less than 11,000 cubic inches. Yay! You can use this tube.Now that you know you can use the tube, you realize that it’s black surface may be uncomfortable in the hot sun since the color black attracts heat. You decide that you will cover it with a white waterproof fabric that you have on hand that you think will be more comfortable.

You realize that the package of fabric says that it is 5,000 in 2. You want to know if you have enough fabric to cover the tube’s surface. Ah-ha! You need to know the tube’s surface area to figure this out. We simply plug our inner and outer radius values into our surface area formula.

We get that the surface area of your tube is approximately 3,148.

4 in 2. Wonderful! You have enough fabric, so you’ll be able to cover the surface of the tube with the fabric for a more comfortable tubing trip!

## Lesson Summary

In mathematics, **torus** is the name we use to describe the 3-dimensional shape of a doughnut. To find the surface area and volume of a torus, we need to know the inner and outer radius of the torus, where the **inner radius** is the radius of the hole in the middle of the torus, and the **outer radius** is the radius of the whole torus. When we have these values, we can calculate the volume and surface area using formulas.

We see that once we have our inner and outer radius measurements, calculating surface area and volume of a torus is a piece of cake, which is great since this shape tends to show up quite a bit in the world around us!