# Lateral Surface Area: Definition & Formula

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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.

Expert Contributor
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.

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 4s^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.

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## Real World Practice Problems for Lateral Surface Area Formulas

### Reminders

• The lateral surface area of an object is equal to the surface area minus the area of the bases of the object.
• The formula for the lateral surface area of a cylinder is 2πrh, where r = radius and h = height.
• The formula for the lateral surface area of a sphere is the same as the formula for the surface area of a sphere, since it has no bases. Thus, the lateral surface area of a sphere is 4πr2, where r = radius.
• The formula for the lateral surface area of a cone is πrs, where r = radius and s = slant height.
• The formula for the lateral surface area of a rectangular box, also called a cuboid, is 2h(l + w), where l = length, w = width, and h = height.

### Practice Problems

1. Joe is putting up a fence around his rectangular garden. The garden has a length of 20 feet, and a width of 18 feet. Joe wants the height of the fence to be 4 feet. What is the minimum amount of fencing material that Joe needs to fence in his garden?
2. Some farmers are erecting a cylindrical silo on their farm that is to be 30 feet tall, and its floor and ceiling have a radius of 10 feet. The sides of the silo will be made of metal, and the floor and ceiling will be made of a special tile. What is the minimum amount of metal that will be needed to build this silo?
3. An ice cream shop serves ice cream in plastic cones shaped like an ice cream cone for individuals that don't want an edible cone. Sharon gets a cone and is curious how much plastic is needed to make these cones. She measures the radius of the cone to be 5 centimeters, and she measures the slant height to be 16 centimeters. What is the minimum amount of plastic needed to make one cone?
4. The Earth is spherical in shape. The radius of the Earth from its center to one of the poles (north or south) is 3,950 miles. The radius of the Earth from its center to the equator is 3,963 miles. What is the range of values for the lateral surface area, or surface area, of the Earth?

### Solutions

1. The minimum amount of fencing needed for Joe's garden is equal to the lateral surface area of a rectangular box with a length of 20 feet, a width of 18 feet, and a height of 4 feet. Thus, we plug these values into the formula for the lateral surface area of a rectangular box, and simplify to get 2h(l + w) = 2(4)(20 + 18) = 304. We get that Joe will need a minimum of 304 square feet of fencing material to fence in his garden.
2. The minimum amount of metal needed to erect the silo is equal to the lateral surface area of a cylinder with a height of 30 feet and a radius of 10 feet. We find this lateral surface area by plugging r = 10 and h = 30 into our formula for the lateral surface area of a cylinder, and simplifying to get rh = 2π(10)(30) = 600π ≈ 1884.96. We get that the minimum amount of metal needed to erect the silo is approximately 600π square feet, or approximately 1884.96 square feet.
3. The minimum amount of plastic needed to make one cone is equal to the lateral surface area of a cone with a radius of 5 centimeters and a slant height of 16 centimeters. We can plug these values into our formula for the lateral surface area of a cone, and simplify to get πrs = π(5)(16) = 80π ≈ 251.33. We get that the minimum amount of plastic needed to make one cone is 80π square centimeters, or approximately 251.33 square centimeters.
4. Using our formula for the surface area of a sphere: if we go with the radius from the the center of the Earth to the north or south pole, then the surface area of the Earth would be r2 = 4π(3950)2 ≈ 196,066,798. If we go with the radius from the center of the Earth to the equator, then the surface area of the Earth would ber2 = 4π(3963)2 ≈ 197,359,488. Therefore, the range of values for the lateral surface area, or surface area, of the Earth is 196,066,798 square miles to 197,359,488 square miles.

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