In this lesson, you will learn the definition of the lateral surface area of a three-dimensional object. You will apply lateral surface area formulas for some three-dimensional shapes to see how to find the lateral surface area of various objects.

## Surface Area

There are a couple of things we need to understand before we define the lateral surface area of an object. Let’s start by defining what the surface area of an object is. The **surface area** of a three-dimensional object is exactly what it sounds like. It refers to how much area the surfaces of the object take up all together. For example, consider a cube. A cube is made of six square sides, also called faces. The surface area of a cube would be the area of each of these six sides added together, or 6 times the area of one of the sides. Let’s look at a six-sided die. The six-sided die pictured here has a side length of 19mm.

Since the side length is 19mm, each side has an area of 19 * 19 = 361 square millimeters. The surface area is all of the six sides added together, so the surface area of a six-sided die with side length 19mm is 361 * 6 = 2,166 square millimeters.

## Base of a Three-Dimensional Object

The next thing we want to take a moment to discuss is the base of a three-dimensional object. The **base** of a three-dimensional object is the bottom side (or face) of the object. When there is a top and a bottom face, both of these are considered to be bases. For instance, our cube has a top and a bottom face. Both of these would be considered to be bases of the cube.

To understand this further, let’s look at some additional three-dimensional shapes and decide how many bases they have.

The first picture is of a rectangular box. We see that this box has a top and a bottom rectangular face, so it has two bases. The second picture is of a cone. Notice that the cone has a bottom circular face, but the top meets at a point, so the cone has only one base. The third picture is of a cylinder. We see that the cylinder has a top and a bottom circular base, so the cylinder has two bases. Lastly, the fourth picture is of a sphere. The sphere is a bit of a special case, because we notice that there is no top or bottom face. Thus, the sphere has no base.

## Lateral Surface Area

Now, let’s talk about **lateral surface area**. The lateral surface area of a three-dimensional object is the surface area of the object minus the area of its bases. For example, consider our die. We found that the surface area of a six-sided die with side length 19mm to be 2,166 square millimeters. To find the lateral surface area of this die, we subtract the area of the two bases. We found that one side of the die has an area of 361 square millimeters. Since the die has two bases, we subtract 361 * 2 from our surface area. That is 2,166 – 361 * 2 = 1,444 square millimeters. Thus, the lateral surface area of our die is 1,444 square millimeters.

In this example, we found the surface area and then subtracted the area of our bases. In many cases, we have a formula for the lateral surface area of an object that simplifies this process. In the case of a cube, the lateral surface area consists of the area of four of the cube’s sides added together, or 4 times the area of one of the cube’s sides. Thus, the lateral surface area of a cube can be found using the formula 4*s*^2, where *s* is the side length of the cube. Considering our die example again, if we plug *s* = 19mm into this formula, we get 4 (19)^2 = 1,444 square millimeters. We see that this is the same answer we got when we found the surface area and then subtracted the area of the two bases of the die. Let’s look at some other formulas for lateral surface area.

## Common Lateral Surface Area Formulas

This chart displays the lateral surface area formulas for the three-dimensional objects we’ve mentioned. You can see that the lateral surface area of a cylinder is 2(pi)rh. The lateral surface area of a sphere is 4(pi)r^2. The lateral surface area of a cone is (pi)rs. And the lateral surface area of a rectangular box is 2h(l+w).

These formulas are all derived by considering the surface area of the object minus the area of the object’s bases. We can use these formulas to find the lateral surface area of many different three-dimensional objects. For example, consider the cardboard portion of a paper towel roll.

Notice that this tube is a cylinder with the bases taken away. Thus, if we find the lateral surface area of this cylinder, we will have found the surface area of this tube. The radius of a paper towel tube is approximately 2.5 centimeters, and the height of the tube is approximately 24 centimeters. From our chart, we see that the lateral surface area formula for a cylinder is given by *L.S.A* = 2(pi)*r**h*. Plugging in our values for the radius and height gives us *L.S.A* = 2(pi)(2.5)(24) = 376.99, when rounding to two decimal places. Therefore, the lateral surface area of our paper towel is approximately 377 square centimeters.

## Lesson Summary

We discussed that the **surface area** of a three-dimensional object is the area of all of its surfaces added together. The **base** of a three-dimensional object is the bottom face, and the top face, when present, of the object. A three-dimensional object can have one, two, or no bases.

When we want to find the **lateral surface area** of a three-dimensional object, we can find the surface area and then subtract the area of its bases or, in many cases, we can use a formula to find the lateral surface area. Anytime we are working with a three-dimensional object and we are interested in the surface area of just its sides, excluding its top and bottom, we are interested in the object’s lateral surface area. We now have the tools to work with this type of surface area efficiently and effectively.

## Key Terms

**Surface area** – how much area the surfaces of the object take up all together

**Base** – the bottom and/or top side (or face) of a three-dimensional object

**Lateral surface area** – the surface area of the object minus the area of its bases

## Learning Outcomes

Studying this lesson and advancing your knowledge of lateral surface area will prepare you to:

- Define surface area
- Identify the base and lateral surface area of a three-dimensional object
- Find the total surface area of a three-dimensional object
- Recognize lateral surface area formulas