Video: Surface Areas and Volumes of Prisms

Apply your knowledge of calculating the surface area and volume of a prism and the area of a regular polygon given a side length to more complex problems involving surface areas and volumes of prisms, including finding side lengths from areas and volumes.

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Video Transcript

In this video, we are going to look at some more advanced problems related to calculating the surface area and the volume of prisms.

So this is the first problem. It says: The surface area of a cube is 294 centimeters squared. We’re then asked to calculate the volume of this cube.

Now in order to calculate the volume of a cube, we would need to know what the length of its sides are. However, we aren’t given that. We’re given the surface area instead. So this problem is going to involve working backwards from knowing the surface area to calculating the side length and then calculating the volume of the cube. So let’s think about how to approach this problem then.

I’ve sketched myself a cube and have decided to call this unknown side length 𝑥. So in order to work out the total surface area, what would have been done is we’d have worked out the area of each of these faces. Now they’re all squares, and as the dimensions are 𝑥, the area of each face will be 𝑥 squared. Now on a cube, there is six faces, six identical faces. So the total surface area of this cube would be six 𝑥 squared. And we’re told in the question that this is equal to 294 centimeters squared. So what this enables me to do is set up an equation involving this unknown side length 𝑥.

So now I need to solve this equation in order to work out what 𝑥 is. Well, the first step will be to divide both sides of this equation by six. And that will give me that 𝑥 squared is equal to 49. And then the next step to work what 𝑥 is, I need to take the square root of both sides of this equation. And that tells me that 𝑥, the side length of this cube, is seven centimeters. Okay, so now I know the side length, it’s relatively straightforward to calculate the volume of this cube. I just need to remember that to calculate the volume of a cube, I multiply these three dimensions together. So I’m doing 𝑥 times 𝑥 times 𝑥 or 𝑥 cubed. So the volume of this cube is seven times seven times seven, which gives me an answer of 343 centimeters cubed or cubic centimeters for the volume of the cube.

So in that question, we saw how to work backwards from knowing the surface area of the cube, calculating the side length, and then calculating the volume. A similar question could perhaps give you the volume and ask you to calculate the surface area.

So here is such a question. It says: The total surface area of the cuboid shown is 86 centimeters squared. And then we’re asked to calculate the volume of this cuboid.

So what you’ll notice looking at the diagram is we’ve been given two of the dimensions, three centimeters and five centimeters. But we’re not told what that third dimension is. So it’s been given the letter 𝑦. In order to calculate the volume, we’re going to need to work out what 𝑦 is. So we’re gonna need to work backwards from knowing the total surface area, calculating this missing dimension 𝑦, and then using it in order to calculate the volume. So let’s think about the different faces of the cuboid that would make up this total surface area.

So we have the front and the back of the cuboid. So we have these faces here. Now if you think about the dimensions of those faces, they are three centimeters and five centimeters. So the area of each of those faces is 15 centimeters squared. Now of course, there are two of them, the front and the back. So we’ll have to remember that later on in our method. Now let’s think about the top and the base of this cuboid, so this face on the top and the one on the bottom. Now the dimensions of that are three centimeters and then this unknown measurement 𝑦, which means the area of those faces. Well, it’s three multiplied by 𝑦. So it’ll be three 𝑦 centimeters squared. Again remember there are two of those.

Finally, let’s think about the sides of this cuboid, so the ones I’m currently shading in orange, and the dimensions of those. Well, the height is five and then the width of this rectangle is again that unknown measurement 𝑦. So it will be five multiplied by 𝑦 or five 𝑦 centimeters squared. So now, we have all the information we need to formulate an equation for this surface area. So we would have added all of these together. So we’d have done three 𝑦 plus five 𝑦 plus 15. But remember there are two of each of those. So we’d also have multiplied that whole expression by two. And then the answer to that we’re told is 86.

So what we’ve done here is formulate an equation for the total surface area in terms of that unknown measurement 𝑦. We now want to solve this equation. So let’s tidy up a little bit first. Three 𝑦 plus five 𝑦 is eight 𝑦 in the brackets. So I have two lots of eight 𝑦 plus 15. Next, I’m gonna divide both sides of this equation by two. So that gives me eight 𝑦 plus 15 is equal to 43. Now, I’m gonna subtract 15 from both sides of this equation. And that would give me eight 𝑦 is equal to 28. The final step in calculating 𝑦 is I need to divide both sides of the equation by eight. And that tells me that 𝑦 is equal to 3.5. So we’ve worked out that this missing dimension of the cuboid is equal to 3.5 centimeters.

Now that I know that, calculating the volume is relatively straightforward. I need to multiply the three dimensions of the cuboid together. So I’m gonna be doing three multiplied by five multiplied by 3.5. And if I then evaluate that, it gives me an answer of 52.5 centimeters cubed. So in this question, we knew the total surface area. In order to calculate the volume, we had to work out the unknown length in this cuboid. We did that by formulating an algebraic equation and then solving it. Once we had that, we could just calculate the volume of the cuboid in the usual way.

Okay, the next question we’re going to look at says, “Tom wants to pour 55 liters of water into the drinking trough shown. Will all of the liquid fit?

So looking at the diagram, we have a drinking trough in the shape of a triangular prism. And we want to know is the capacity of that trough, is it more or less than 55 liters. So we’re going to need to work out the volume of this trough. Now there’s something we need to be careful of, which is if you look at the diagram, the units that we’ve been given are mixed: two of the measurements are centimeters and one of the units is meters. So we need to make sure that they’re all in the same unit before we start trying to calculate this volume.

So the simplest thing to do then is to convert this 1.5 meters into 150 centimeters. And we’ll use that in our calculations. So we’re going to calculate the volume of this triangular prism, this trough. Now that answer will be in centimeters cubed. So we’ll have to think later about how that relates to liters, because the information in the question is about 55 liters of water. So for the volume of this triangular prism then, remember we need to work out the area of the cross section. So that’s this triangle here. And then multiply it by the height or the depth of the prism. So that’s the 150 centimeters.

Now the cross section is a triangle. So we’re going to be doing base times perpendicular height over two. So that’s 35 times 20 over two. And then, we’re going to multiply by the depth of the prism. So that’s the 150 centimeters. So this gives us a volume of 52500 centimeters cubed for the drinking trough. So we need to think how this relates to liters. And we need to recall a key piece of information, which is that one centimeter cubed is equivalent to one milliliter. So what this tells us then is that the volume of this trough in terms of how much liquid can be poured into it. Well, it’s a one-for-one conversion between centimeters cubed and milliliters. So we can fit 52500 milliliters of water in this trough.

Now the information in the question is given in terms of liters. So let’s convert this to liters by dividing by 1000. And this tells us then that the volume of this trough is 52.5 liters. So Tom wants to pour 55 liters in there. Well, he’s not gonna be able to. He’s gonna have some liquid that overflows from the trough. So an answer to the question will all the liquid fit, no it won’t. There’s too much liquid to fit in that drinking trough.

Okay, the final question that we’re going to look at: we’re given a right regular octagonal prism. And we’re asked to find the surface area of this prism.

So first of all, let’s just think about the different faces that we need to consider. Well, the top and the base of this prism are both octagons. So we need to find the area of those octagons. And then, it also has eight rectangular faces around the edges. So we’re looking for eight rectangles and two octagons. So let’s start off by thinking about these octagons. They’re regular octagons.

And you need to recall from previous work that there is a formula for calculating the area of a regular polygon, and is this formula here that the area of a regular 𝑛-sided polygon is equal to 𝑛𝑥 squared over four multiplied by cot of 180 over 𝑛, where 𝑥 is representing the side length of this polygon and cot remember is equal to one over tan. So for our octagon, the number of sides is eight. So we’re gonna have 𝑛 equals eight. And the side length is three. So we’re going to substitute 𝑥 is equal to three. So the area of each of these rectangles is gonna be eight multiplied by three squared all over four multiplied by cot of 180 over eight. That will give me one of these octagons. But remember they are the same on the top and the base. Therefore, I need to double this in order to get the total area of the two.

Now if I evaluate this using my calculator, I get an answer of 86.9116 for the area of those two together. Now remember I’ve expressed the angle in terms of degrees here because it’s 180. So I need to make sure my calculator is in degree mode when I’m typing this in.

Okay, next we want to work out the area of the rectangular faces, all the sides of this octagonal prism. So they are just rectangles and their dimensions are the three centimeters which is the side length of the octagon and 10 centimeters which is the height of the prism. So each of these faces will have an area of 10 times three. But remember there’re eight of them. So I need to multiply that by eight in order to work out the total. So that gives me 240 as a contribution to the area from the sides.

So I’ve now got all the different parts of the area that I need. So to work out the total surface area, I just need to add these together, so that value of 86.9116 plus that value of 240. And so, that gives me a final answer then of 326.9 centimeters squared for the surface area of this prism. Remember the question asked me to find it to the nearest tenth. So that’s how I’ve rounded my answer.

So to summarize then, in this video, we’ve just looked at how to approach a couple of more complex problems related to the surface area and the volume of prisms.

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