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What is a Cuboid Shape? - Definition, Area & Properties

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  • 0:00 What Is a Cuboid Shape?
  • 1:09 Volume
  • 2:49 Surface Area
  • 4:52 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.

The cuboid is one of the most common shapes in the environment around us. In this lesson, we will learn what a cuboid shape is and how to find its volume and surface area.

What Is a Cuboid Shape?

Take a minute to look around the room. Do you see anything that is the shape of a rectangular box? This could be a piece of furniture, a box, a brick, or anything of the like. This shape is called a cuboid, and it is an extremely common shape to see in the world around us. This is demonstrated in the following picture of a building, a book, and an ice cream sandwich. All of these objects are cuboid shapes.

Examples of Cuboids
cuboid examples

A cuboid is a three-dimensional shape with a length, width, and a height. The cuboid shape has six sides, called faces. Each face of a cuboid is a rectangle, and all of a cuboid's corners (called vertices) are 90-degree angles. Ultimately, a cuboid has the shape of a rectangular box.

Cuboid
cuboid

This may leave you wondering what about a cube, or a box with six square faces all the same size. To address this, we need to realize that a square is a special type of rectangle. Therefore, a cube still has faces that are rectangles, and we see that a cube is still a cuboid. Anything that is a box-shaped object is a cuboid.

Volume

The volume of a three-dimensional object is how much space is inside the object. Some of you may be familiar with the volume of a rectangular box. If so, then you are also familiar with the volume of a cuboid. To find the volume of a cuboid, we multiply the cuboid's length times its width times its height. In other words, the volume V of a cuboid is found by the formula V = lwh, where l = length, w = width, and h = height.

Volume of a Cuboid
volume of cuboid

This is a very useful formula. For instance, consider two of our initial cuboid examples, the book and the ice cream sandwich. The volume of the book tells us how much space the pages of the book take up. The volume of the ice cream sandwich tells us how much ice cream is in the sandwich - and let's face it, the ice cream is the best part!

The book shown has a length of 6 inches, a width of 4 inches, and a height of 1 inch. Therefore, we find the volume of the book by plugging l = 6, w = 4, and h = 1 into our volume formula to get V = 6 x 4 x 1 = 24 cubic inches. This tells us that the pages of the book take up 24 cubic inches.

Similarly, the ice cream sandwich has a length of 12 centimeters, a width of 5 centimeters, and a height of 2.4 centimeters. Plugging these values into our volume formula gives V = 12 x 5 x 2.4 = 144 cubic centimeters. This tells us that there is 144 cubic centimeters of ice cream in our ice cream sandwich - yum!

Surface Area

The surface area of a three-dimensional object is the area of all of the object's sides added together. That is, it is the total area of all of the object's surfaces. To find the surface area of a cuboid, we want to add up the areas of each of the cuboid's faces. Thus, the surface area of a cuboid is:

SA = (Area face 1) + (Area face 2) + (Area face 3) + (Area face 4) + (Area face 5) + (Area face 6)

Notice that in a cuboid, parallel faces are the same, so they have the same area. Thus, the formula becomes:

SA = 2(Area face 1) + 2(Area face 2) + 2(Area face 3)

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