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High School Geometry: Help and Review13 chapters | 162 lessons

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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.

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.

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.

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.

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.

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!

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)

The cuboid's faces are rectangles, so each of their areas is found by multiplying the face's length times its width. Therefore, the formula becomes:

*SA* = 2*lh* + 2*wh* + 2*lw*

Where *l* = length of the cuboid, *w* = width of the cuboid, and *h* = height of the cuboid.

This formula is useful when we want to know the area of the surface of a cuboid. Consider our initial examples again; namely, consider the building. If we assume the bottom and top of the building are made from the same material that the sides are, then the surface area of the building would tell us how much material was needed to create the outside of the building.

The building has an approximate length and width of 340 feet and an approximate height of 860 feet. To find the surface area of the building, we plug *l* = 340, *w* = 340, and *h* = 860 into our surface area formula to get *SA* = (2 x 340 x 860) + (2 x 340 x 860) + (2 x 340 x 340) = 1,400,800 square feet. Thus, it took 1,400,800 square feet of material just to create the outside of that building.

Let's review. A **cuboid** is a box-shaped object. The **volume of a cuboid** is given by the formula *V* = *lwh*, and the **surface area of a cuboid** is given by the formula *SA* = 2*lh* + 2*wh* + 2*lw* where *l* = length, *w* = width, and *h* = height. Because cuboids show up so often in the world around us, it is extremely useful to be familiar with these shapes and their properties. The tools we've learned in this lesson allow us to do just that.

- Three-dimensional shape that has a length, width, and height
- Box-shaped
- Has six rectangular sides called faces
- All corners form 90-degree angles

As you complete this lesson on cuboids, you might display the ability to:

- Illustrate a cuboid
- Calculate the volume and surface area of a cuboid
- Recognize the formulas for finding a cuboid's volume and surface

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