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Lateral Surface Area: Definition & Formula

Lateral Surface Area: Definition & Formula
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  • 0:00 Surface Area
  • 1:05 Base Of A…
  • 2:15 Lateral Surface Area
  • 4:05 Common Lateral Surface…
  • 5:40 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

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.

Six-Sided Die
six sided die

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.

Bases of a Cube
cube bases

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

Bases of Three-Dimensional Objects
bases

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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