Video: Surface Areas of Rectangular Prism and Cubes

In this video, we will learn how to calculate surface areas of rectangular prisms and cubes and use this to solve real-life problems.

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Video Transcript

In this video, we’ll learn how to calculate surface areas of rectangular prisms and cubes and use this to solve real-life problems. We’ll begin by recalling what a rectangular prism, sometimes called a cuboid, and a cube actually look like. And how their properties can help us to find the surface area, which is just the total area of all their faces. We’ll also learn how we can use the information about the surface area of a cube to calculate missing dimensions.

So our first question is, what is a prism? A prism is a three-dimensional shape with a constant cross section. In other words, the cross section has the same shape and size throughout its length. A triangular prism, for example, has a triangular cross section. I could slice the shape down here or down here and the size and the shape of that triangle would stay the same. Similarly, a cylinder has a constant cross section which is the shape of a circle. Now, in this video, we’re interested purely in rectangular prisms, like this guy here, and cubes.

A cube is simply a rectangular prism whose dimensions are all the same. We notice that the faces, that’s the flat surfaces, of a cube, in fact, are all squares. So what else do we know about rectangular prisms and cubes? Well, let’s imagine now that we’re breaking a rectangular prism down into its net. The net might look a little like this. We can see this net has one, two, three, four, five, well, six faces. This is probably the most important fact about rectangular prisms and indeed cubes when it comes to calculating the surface area. That’s the sum total of the areas of all of the faces. To find the surface area of a rectangular prism or a cube then, we simply calculate the area of all six faces and then find their sum. Let’s see what that might look like.

Find the surface area of the rectangular prism shown.

In this question, we have a rectangular prism. It’s so called because its cross section is a rectangle. But it so happens that all of its faces are also rectangles. Now, the question is asking us to find the surface area of the prism. Well, the surface area is the total area of all of its faces. And so we begin by recalling that rectangular prisms, sometimes called cuboids, have exactly six faces. Our job then is to identify each of these faces and calculate their area. So let’s identify one of the faces of the rectangular prism.

What about this face at the front of our shape? It’s quite clearly a rectangle, and we could see that the length of its base is 29 meters. We also know that the area of a rectangle is its base multiplied by its height. So what is the height of this rectangle? We do know that the height of our prism is 15 meters, so this dimension here must also be 15 meters. That’s the height of our rectangle. And therefore, the area of this rectangle is 29 multiplied by 15. We have a number of ways to calculate this. We could use the column method or the grid method or even a calculator. Let’s look at the column method.

We begin by multiplying nine by five to give us 45. We place the five in the units column and then carry that four. We then calculate two multiplied by five, which is 10. We’re going to add that four, giving us 14. Now, it might feel like the next sum we’re going to do is nine multiplied by one. But in fact, we’re technically multiplying nine by 10. So we add a zero here. Nine multiplied by one is nine. Two multiplied by one is two. And all that’s left is to add these two numbers. When we do, we find that 29 multiplied by 15 is 435. Now, since we’re working in meters, the area of this front face that we’ve highlighted in yellow is 435 square meters.

Now, in fact, if we consider the shape of our prism, we know that there’s another face identical to this. It will be at the back of the shape. And so we have two rectangles with an area of 435 square meters. So, we can add this measurement here, or we could alternatively multiply 435 by two. Let’s find another face. We’ll move on to the face at the side of the shape. We know that the length of the base of this rectangle is 14 meters and its height is 15 meters. And so using the formula the area of a rectangle is base times height, the area of this rectangle is 14 times 15.

This time, let’s calculate this using the grid method. We split 14 and 15 up into their tens and units. 10 multiplied by 10 is 100, four multiplied by 10 is 40, five multiplied by 10 is 50, and five multiplied by four is 20. 14 multiplied by 15 is the sum total of these four values. That’s 210 or 210 square meters. Once again, we know that the face that is parallel to this in our prism, that’s back here, will have the same area. And so we add another measurement of 210 square meters. So far, we’ve calculated the area of four faces. We have two more to find.

Let’s consider the rectangular face at the top of our diagram. We know that this line here is parallel to this one, so this dimension must be 29 meters. Similarly, this dimension is the same as this one. It’s 14 meters. And so the area of the face at the top of our prism is 29 multiplied by 14, which is 406 square meters. And we know that the face that is parallel to this one, that’s the one at the bottom, must have the same area. That’s another measurement of 406 square meters. We now have one, two, three, four, five, six measurements as we required.

Remember, the surface area is the total area of all our faces, so we’re going to finish by adding all of these values. The sum total of these six values is 2102, and so the surface area of a rectangular prism is 2102 square meters. Now, units are really important. And a common mistake here is to use a cubic unit of measurement, such as meters cubed or centimeters cubed. Remember, though, when we’re working with areas, our units are squared. So here, we work in square meters.

We’ll now consider a similar example. But this time, we’re going to look at calculating the surface area of a cube.

Find the surface area of a cube of length 11 centimeters.

In this question, we’ve been given some information about a cube. We recall that a cube is a rectangular prism whose edges are all equal in length. In fact, it’s a rectangular prism whose faces are all squares. We also know that the surface area is calculated by adding together the areas of all of the three-dimensional shapes’ faces. And we know the rectangular prisms and cubes all have six faces. This means if we can find the area of one of the faces of our cube, we can find the total surface area by multiplying this by the number of faces, by six. These squares are 11 centimeters by 11 centimeters. And the area of a square is found by multiplying the length of its base by its height or, alternatively, by squaring the dimension of the base.

In this case then, the area of one face, the area of one of our squares, is 11 times 11. That’s 121. Now, we’re working in centimeters, so the units here are square centimeters. We can therefore say that the total surface area of our cube is six multiplied by 121. Let’s use the column. We work out one multiplied by six, which is six. Two multiplied by six is 12, so we put a two here and carry the one. Then one multiplied by six again is six. When we add the one, we get seven. This means six multiplied by 121 is 726. And therefore, the surface area of the cube with a length of 11 centimeters is 726 square centimeters.

Now, finding the surface area of prisms given a real-world context can make this scenario a little bit more tricky. Let’s see what that might look like.

Suppose the length of each edge of an ice cube is 19 centimeters. The cube is then cut horizontally in half into two smaller rectangular prisms. Determine the surface area of one of the two prisms.

In this question, we’ve been given some information about a cube. And we recall that a cube is a rectangular prism whose edges are all equal in length. In fact, it’s a rectangular prism whose faces are all squares. In this case, each dimension is 19 centimeters. The problem is, this is not the shape we want to find the surface area of. The shape is cut horizontally in half. And that leaves us with two identical or congruent rectangular prisms or cuboids. So what are the dimensions of each of these rectangular prisms?

Well, we know that two of the dimensions remain 19 centimeters. The third dimension, however, has been halved; it’s going to be 19 divided by two centimeters. That’s a calculation we might be able to perform in our head, or we can use the bus stop method. We say, how many twos make one? Well, that’s zero. So instead, we say, how many twos make 19? That’s nine, but we have a remainder of one. So where does this remainder go? We add a decimal point and a zero. Remember, 19 and 19.0 are the same number. We carry our decimal point up here. And we now ask ourselves, how many twos make 10? Well, that’s five. And so we see that the third dimension of our rectangular prism is 9.5 centimeters.

We want to calculate its surface area. That’s the sum total of the areas of all of its faces. And in fact, we recall that a rectangular prism has six faces, so we’re going to need to find six area measurements. We’ll begin by considering the front face of a rectangular prism. We know that the area of a rectangle is the length of its base multiplied by its height. The base of this rectangle is 19 centimeters. Its height is the length of this edge. Well, this is parallel to this edge, so it’s 9.5 centimeters. And so the area of this first rectangle is 19 multiplied by 9.5, which is 180.5. Now, we’re working in centimeters, so that’s 180.5 square centimeters.

There’s another face exactly like this one, and it sits at the back of our prism. And so the area of the two faces combined must be two times 180.5. That’s 361 square centimeters. Let’s now move on to the area of this face. Once again, its area is 19 multiplied by 9.5. So again, we have 180.5 square centimeters. There’s an identical face to this that sits at the back of our shape here. And so we can double this measurement again to get 361 square centimeters. Notice that, alternatively, we could’ve multiplied 180.5 by four in the first place. We’ve calculated the area of four faces. We need two more. Let’s look at the face at the top and, in fact, the identical one at the bottom of our shape.

This time, its dimensions are 19 centimeters and 19 centimeters. Remember, it’s that square face from earlier. 19 multiplied by 19 is 361. And since there’s two of these, we multiply this by two. And we see that the combined area of the top face and bottom face is 722 square centimeters. The surface area is the sum total of these values. It’s 361 plus 361 plus 722, which is 1444 square centimeters. Now, a common mistake here would have been to calculate the surface area of the cube and then halve it. This would not give us a correct answer as we haven’t halved each face, just four out of six of them.

We’re now going to consider how information about the perimeter of a base of a cube can help us find its surface area.

If the perimeter of the base of a cube is 54.4 centimeters, find its total surface area.

We begin by recalling that a cube is a three-dimensional shape whose faces are all squares. We’re told some information about the perimeter of one of these squares. It’s 54.4 centimeters. Now, we know that the perimeter is the total distance around the shape, and we also know that all sides in a square are equal. Now, since a square has four sides, we can calculate the side length of our square by dividing 54.4 by four. And when we do, we find the side length of our square to be 13.6 centimeters. The question wants us to find the total surface area of the cube. We now know that all the dimensions of the cube are 13.6 centimeters, and the surface area of a three-dimensional shape is the sum total of all of the areas of its faces.

We begin then by calculating the area of one of the faces on our cube. Well, the area of a square is base times height or its side length squared. That’s 13.6 times 13.6, which is 184.96 square centimeters. Remember, a cube has six identical faces, so the surface area is the area of one of its faces multiplied by six. That’s six times 184.96, which gives us a total surface area of 1109.76 square centimeters.

In our very final example, we’ll look at how we can calculate dimensions of a cube given its surface area.

The surface area of a cube is 1020 square centimeters. What is the area of one face of the cube?

Remember, the surface area of a three-dimensional shape is the total area of all of its faces. We’re looking to find the surface area of a cube. Now, we know a cube is a three-dimensional shape whose faces are all squares. It has six identical faces. Since all of these faces will have the exact same area, we can calculate the area of one of the faces by dividing the surface area by six. We might use the bus stop method to perform this calculation. We say, how many sixes make one? Well, it’s zero. And instead, we ask ourself, how many sixes make 10? It’s one remainder four.

Next, we ask ourself, how many sixes make 42? That’s seven. And how many sixes make zero? That’s zero. This means that 1020 divided by six is 170. The surface area measurement is given in square centimeters. Now, we’re still looking at areas, so our units are also square centimeters. And we can therefore say that the area of one face of our cube is 170 square centimeters.

In this video, we’ve learned that a prism is a three-dimensional shape with a constant cross section. We saw that a rectangular prism in particular has a rectangular cross section but also six rectangular faces and that a cube is a rectangular prism whose dimensions are all equal, whose faces are all squares. We learned that we can calculate the surface area of a rectangular prism by adding together the area of all six faces. And that it’s important to remember that when we’re working with area and surface area, we need to work with squared units, such as centimeters squared and meters squared.

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