Video Transcript
In this video, we’re going to look at how to calculate the surface area of a prism.
Now, a prism remember is a particular type of three-dimensional shape. It has the special property that if you cut it at any point along its length, the face that you see is always constant. It has what’s referred to as a constant cross section. So, the examples on the screen here, there’s a cube. If you cut that at any point in its length, you will always see a square. A cuboid, if you cut that at any point, you will see a rectangle. And the triangular prism, if you cut that any point, you will see this triangular face here.
Now, what’s meant by the surface area of a prism is the total area of all of the prism’s faces. So, you can also think of this in a couple of different ways. The first perhaps as you can imagine, if you were to wrap up one of these prisms, then it’ll be the exact area of wrapping paper that you would need in order to cover all of its faces with no overlaps and no gaps.
Or you can think of it as if you were to unwrap one of these prisms and make it two-dimensional, then it’s the exact area of card, for example, that you would need in order to create what we call the net of the prism. So, the net is that the two-dimensional template that you would create in order to then fold it up to build one of these three-dimensional prisms. So, we’ll see how to calculate the surface area prism for a couple of different types.
So, our first question asks us to find the surface area of a cube with side length eight centimeters.
Now, we’ll answer this question by thinking about the net of a cube. So, we need to think about what are the different faces that this cube has. Now, a cube has six faces. And they’re all squares with side length of eight centimeters. Now, there are lots of different ways that you can arrange these six squares when you draw out the net of a cube, but perhaps the most common one is to see them arranged in a sort of cross shape, such as this one here. If you wanted, you could draw out a net like this on a piece of card or paper and cut it out and then fold it up and check that it does in fact fold back into that cube shape.
So, now, we know what the net of the cube looks like. We just need to work out what the area of this card would be. So, we’ve got six faces. And in the case of the cube, they’re all exactly the same. So, we can just work out the area of one and then multiply it by six. Now, each of these faces is a square and the side length is eight centimeters, so the area of a square, we’re gonna multiply eight by eight and that will give us 64 centimeters squared. So, that’s the area of each of these square faces.
To work out the total surface area then, we need to multiply this 64 by six. And then, that will give us an answer of 384 centimeters squared. So, drawing out the net of whichever prism we’re interested in is a really useful way of making sure we can visualize all of the different faces that are involved and make sure we include the areas of all of the faces in our calculations. Just a note about units, we are looking at 3D shapes, but we’re thinking about unfolding them into two-dimensional shapes. Therefore, the units are area units, centimeters squared, millimeters squared, and so on.
Our next question is about a cuboid, or a rectangular prism. And it asks us, find the surface area of this cuboid below, which has dimensions of five, 12, and three millimeters.
So, we could answer it like we did the question of the cube. We could draw out the net of this cuboid. But I’m actually going to approach this one in a slightly different way. Rather than drawing out all of the faces, I’m just going to picture the different faces that are involved. So, a cuboid also has six faces, but unlike the cube, they’re not all the same because the measurements are different. However, they do come in pairs. And there are three pairs of faces that we need to think about, the front and the back, which are the same as each other, the top and the base, and then the left and the right.
So, I’m going to break this calculation down into three stages where I find the areas of those three pairs. So, I’m going to start with the front and the back first of all. The front and the back are rectangles and they have measurements of five millimeters and then three millimeters for the height. So, as they’re rectangles, the areas for those are just gonna be found by multiplying the three and the five together. And as there are two of them, I’m going to include a factor of two as well. So, this gives me a contribution of 30 millimeters squared for the front and the back.
Next, I’m gonna think about the top and the base of the cuboid, which are these faces that I’ve marked in green. Now, they’re rectangles. And the measurements for the top and the base are five millimeters and 12 millimeters, so to multiply those two together. Then again, it’s the top and the base. I have two of them, so I also need to multiply that by two. So, that gives me a contribution of 120 millimeters squared for the top and the base.
Lastly, I need to think about the sides of the cuboid, so the left and the right. These are also rectangles, and their measurements are 12 and three millimeters. So, I’m gonna multiply 12 by three. And again, as in the case of all the other pairs, there are two of them. So, I also need to double that. And so, that gives me 72 millimeters squared for the two sides.
Final step then, I want the total surface area. So, I need to add together these areas that I’ve worked out, so 30 plus 120 plus 72. And that gives me a total surface area for the whole cuboid of 222 millimeters squared. So, a quick recap of what we did there, we looked at the three different pairs of faces that we had, the front and the back, the top and the base, and then the two sides. And then, we worked out those areas and added them all together. So, we had six faces included in the calculation overall.
Our final question asks us to find the surface area of the triangular prism shown.
So, looking at the diagram, we’ve got a triangular prism. And it is a right-angled triangle. We can see that from the label in the diagram. So, you could draw out a full net for the triangular prism or you could just carefully think about what are the different faces that are involved. Now, you might want to take a minute just to pause the video and visualize how many faces there are and what shapes each of them are.
So, there are, in fact, five faces on a triangular prism. There are the right-angled triangles that we see at the front and the back, which are the same as each other. There’s a rectangle on the flat base of this prism. There’s another rectangle which is this sort of face that we can’t see at the side of the prism. And then, there’s the sloping side, which is also another rectangle. And we’ll need to think carefully about the dimensions of each of these.
So, let’s start off with the base. The base as I said is a rectangle. And the dimensions of the base are three meters and seven meters. So, to find the area of this rectangular base, we’re just gonna multiply the three and the seven together. So, that gives us 21 meters squared as the contribution from the base.
Now, let’s think about the triangular faces on the front and the back of this prism. So, for a triangle, we do the base multiplied by the perpendicular height and then divide it by two. So, for these triangles, that will be three times four divided by two. But then, as there are two of them, the front and the back, we also need to multiply this by two. So, that division and multiplication by two actually cancels itself out. And we’re left with 12 meters squared as the contribution from the front and the back.
Now, let’s think about the vertical face behind this prism. So, that’s the face sort of round the side here that we can’t really see. So, this is also a rectangle. And it’s got a height of four here. And then, this dimension here is seven meters. So, the area of this vertical face then, it’s a rectangle, so four times seven, it’s gonna be 28 meters squared.
So, that accounts for four of the five faces. And the last one that we need to think about is this sloping face here, the one that I’m marking in in purple at the moment. Now, this is also a rectangle. And we can see that one of its dimensions is seven meters. But we need to think carefully about what its other dimension is, so what this length here is. And in order to do that, we need to look at the right-angled triangle that this length is part of, so the right-angled triangle on the front of this triangular prism.
Now, you need to recall some other work within mathematics. You need to recall the Pythagorean theorem, which tells us how the lengths in a right-angled triangle are related to each other. And if you remember, the Pythagorean theorem tells us that, in a right-angled triangle, the square of the hypotenuse, the longest side, so that is this red side here, is equal to the sum of the squares of the two shorter sides. So, in this triangle, that’s the three meters and the four meters.
So, I need to do a little bit of working out. If I give this a letter, perhaps the letter 𝑥, then I can write down what the Pythagorean theorem tells me for this triangle. So, it tells me that 𝑥 squared is equal to three squared plus four squared. Then, I can go through the working out to evaluate what 𝑥 squared is equal to. So, three squared and four squared are nine and 16, and when I add them together, I get 25. If I then take the square root, that tells me that 𝑥 is equal to five.
Now, you may have been able to spot that without actually doing any working out because three-four-five is an example of a Pythagorean triple. That is, a right-angled triangle where the lengths of the three sides are all integers. So, if you knew that, you may have been able to take a little shortcut there. At anyway, we’ve worked out that the length of this side is five meters. So, now, you just need to work out the area of this final face.
So, I just work it out at the bottom here. These little sketches are not to scale by the way. So, sloping face with measurements of seven meters and five meters, so its area will be seven multiplied by five, so 35 meters squared for the area of that final sloping face. Right, the final step in this calculation then is I just need to add together the areas of all of these faces. So, the total surface area then, 21 plus 12 plus 28 plus 35, which gives me a final answer of 96 meters squared, or square meters.
So, to summarize then, the surface area of a prism is the total area of all of its faces. You need to make sure you account for all of the faces. And you can either do that by drawing out the net of the prism. Or you can just visualize the different faces and the dimensions. Or perhaps you could sketch them all out separately like we did in this example here.
