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
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.
Volume of a 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!
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)
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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 = 2lh + 2wh + 2lw
Where l = length of the cuboid, w = width of the cuboid, and h = height of the cuboid.
Surface area of a 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 = 2lh + 2wh + 2lw 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.
Properties of Cuboids
All of these objects are cuboids.
Three-dimensional shape that has a length, width, and height
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
Suppose Greg wants to wrap a Christmas gift that is in a rectangular box. He is trying to determine how much wrapping paper to buy in order to wrap the box completely. He takes some measurements of the box, and finds that it has a length of 7 inches, a width of 5 inches, and a height of 4 inches. Based on these measurements, what is the minimum amount of wrapping paper Greg needs to buy to wrap the box?
Marisa received flowers for her birthday. She has a rectangular vase with a length of 3 inches, a width of 6 inches, and a height of 8 inches. She fills one of the vases with water so that the height of the water is even with the height of the vase. When she places the flowers in the vase, a lot of the water spills out. When she takes the flowers out of the vase to clean up, she notices that the height of the remaining water is 5.5 inches. How much water spilled out of the vase? Based on this, how much volume, or space, do the flowers take up in the vase?
A construction crew is erecting a building that has a rectangular shape. They want the two ends to be squares, meaning they will have the same length and width. The bottom, top, and two sides are rectangles with a length of 100 feet and a width of 70 feet. Based on this, what are the possible dimensions of the square ends?
Solution to problem 1:
The box Greg wants to wrap is rectangular, so it is a cuboid. Since Greg wants to wrap the outside of the box, he will need enough wrapping paper to cover the surface area of the box. We are given that the length is 7 inches, the width is 5 inches, and the height is 4 inches. Thus, we plug l = 7, w = 5, and h = 4 into our surface area formula for a cuboid, SA = 2lh + 2wh + 2lw, and simplify.
We get that Greg will need to buy 166 square inches of wrapping paper in order to wrap his gift.
Solution to problem 2:
Since the vase is rectangular, it is a cuboid. To determine the amount of water that spilled out of the vase, Vspill, we need to find the volume of the water in a full vase, Vbefore, and the volume of the water after the water spilled out, Vafter. Then, we will find the difference between these two volumes, which will be equal to the amount of water displaced. The full vase has a length of 3 inches, a width of 6 inches, and a height of 8 inches. We plug these into our volume formula for a cuboid, V = lwh.
Vbefore = (3)(6)(8) = 144
To find the volume of the water after the water spilled out, we use our formula again, but this time, we use the new height of the water, which is 5.5 inches.
Vafter = (3)(6)(5.5) = 99
Now, we simply find the difference by subtracting 99 from 144.
Vspill = 144 - 99 = 45
We get that the amount of water that spilled out of the vase is 45 cubic inches. Since the flowers displaced this much water, it must be the case that the amount of volume that the flowers take up in the vase is 45 cubic inches.
Solution to problem 3:
Since the building is a rectangular shape, it is a cuboid, so it has six rectangular sides, two of which are the square ends. These two square ends will either have to have side lengths that are equal to the length or that are equal to the width of the other four rectangular sides. Therefore, the square ends can either be 100 feet by 100 feet, or they can be 70 feet by 70 feet.
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