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Video: Volumes of Similar Solids

Lauren McNaughten

Identify the relationship between the volumes of similar solids. Use the length ratio to calculate the volume ratio and vice versa. Apply this method to calculate the volume of a similar solid or a missing length based on knowing the two volumes.


Video Transcript

In this video, we’re going to look at similar solids and in particular the relationship between their volumes. First of all, let’s clarify what is meant by the term similar solids.

Two solids are similar if, first of all, they have to be the same shape as each other and, secondly, corresponding lengths must be in the same ratio. So for example in the case of a cuboid, if you doubled the width of the cuboid, you must also double the length and the height in order to create two similar solids. You can’t multiply one dimension by two, but multiply another by three for example.

So here as an example are two similar cuboids. If you look at the lengths of the cuboids, you’ll see that they’re always double in cuboid B what they were in cuboid A. So the two centimetres now is four; the three centimetres is now six; and the five centimetres is now ten centimetres.

Now we refer to this as the ratio of lengths or the length ratio between the two solids. And if I take any pair of corresponding sides, so for example the three to six, then their length ratio will be three to six but of course that ratio could be simplified then to one to two. And that ratio is the same for any corresponding pair of sides.

Now we wanted to look at the volumes of these similar solids. So let’s look at the volume of each of these cuboids. And remember to calculate the volume of a cuboid essentially, you just multiply its three dimensions together. So in the case of cuboid A, that will be two times three times five, which gives us thirty centimetres cubed.

And in the case of cuboid B, four times six times ten, which is two hundred and forty centimetres cubed. Now as we did with the lengths, let’s write a ratio for these two volumes. So it will be thirty to two hundred and forty, but that simplifies because both sides of that ratio can be divided by thirty. And so that gives us a volume ratio of one to eight.

Now there’s a key point here, which is that there is a relationship between the length ratio and the volume ratio. It may not be so obvious in the case of the one. But in the case of the two and eight, the relationship is that two cubed is eight. And of course now that you think about it, one cubed is also one.

This is not a coincidence. It’s always the case that whatever the length ratio is, the volume ratio can be found by cubing each of the two sides. So this is illustrative of a general rule that we’re going to use throughout this video, which is that if the length ratio between two similar solids is 𝑎 to 𝑏, then the volume ratio between them is 𝑎 cubed to 𝑏 cubed.

So let’s look at a couple of questions related to this. The first says that solids R and S are similar. Their lengths are in the ratio two to seven. We’re asked what is the ratio of their volumes. So this first question is just a case of applying the general rule that we saw on the previous slide.

And let me remind you that that rule was this: if the length ratio between two similar solids is 𝑎 to 𝑏, then the volume ratio is 𝑎 cubed to 𝑏 cubed. So we can calculate the volume ratio for these solids R and S just by cubing both sides of their length ratio.

So their volume ratio will be two cubed to seven cubed. And evaluating both sides gives us that the volume ratio is eight to three hundred and forty-three.

Now that’s a straightforward question, but it’s just important because people often forget that this relationship exists, and they assume that if the lengths are in a particular ratio then the volumes must also be in that same ratio. But as we saw with the cuboids, that isn’t the case. Right, this is the next question. Cones A and B are similar. Calculate the volume of cone B.

So if you look at the information in the question, you’ll see that we’re given a length for cone A and a length for cone B. They’re both the radius of the circle on the base, and we’re also told the volume of cone A. And we need to use all of this information together to calculate the volume of cone B. So to start off this question then, we’re given corresponding lengths in the two cones, which means we can write down the ratio of the lengths.

So the length ratio between the two cone is three to six, but of course that will simplify to one to two by dividing both sides by three. From this we can then calculate the volume ratio, because remember we saw that in order to calculate the volume ratio from the length ratio, we need to cube both parts of it. So the volume ratio is one cubed to two cubed, which of course is just one to eight.

What this means then is that the volume of cone B is eight times larger than the volume of cone A. So I can calculate it by taking the volume of cone A, which is seventy centimetres cubed, and multiplying it by eight. And doing so it gives me an answer then of five hundred and sixty cubic centimetres for the volume of cone B.

So the important thing to note in this question is it we didn’t have to use a formula for calculating the volume of a cone, even though of course such formula does exist. We just used the relationship between the volumes of the two cone due to the similarity of them.

The final question then that we’re going to look at says the two pyramids below are similar. Calculate the height of the smaller one. So looking at the information in the question, we can see that we’ve been given the volume of both pyramids, and we’ve been given the height of the larger one but the height of the smaller one is missing. And that’s what they’re looking to work out.

So let’s think about what we can write down to start off with. We know what the two volumes are so we can write down a volume ratio between the two pyramids. So the volume ratio is sixteen to fifty-four, and we can simplify that by dividing both sides of the ratio by two. So the simplified ratio is eight to twenty-seven.

Now looking to work out the height of the smaller pyramid, which is a length, so we’d also like to know what the length ratio is. And we can work it out from the volume ratio. Remember that general rule we had that said whatever the length ratio is the volume ratio is the cube of it? This means then that we can work backwards from knowing the volume ratio to calculate the length ratio. But instead of cubing, we will be cube rooting as we’re going back the other way.

So the length ratio will be the cubed root of eight to the cubed root of twenty-seven. And evaluating both of those tells me that the length ratio is two to three. So finally we need to use this ratio to calculate the missing height. Now I’m gonna give it letter; I’m going to call it ℎ.

So what this tells me then is that if I do ℎ, the height of the smaller pyramid, divided by six which is the height of a larger pyramid, this will give me two divided by three using that length ratio, So I have an equation, quite a straightforward one, for this unknown letter ℎ. What I need to do now is solve this equation. So if I multiply both sides of the equation by six, so this turns into ℎ is equal to six multiplied by two-thirds, which is four.

Therefore the height of the smaller pyramid must be equal to four metres. So this question was a little bit more complex than the ones we’ve seen before because it involved working backwards and we had to cube root to the volume ratio in order to get back to the length ratio.

So in summary then, we’ve seen this key relationship that exists between the volumes of similar solids, and we’ve seen how to use it in order to answer questions about calculating the volume of either a smaller or a larger solid and also calculating a missing length in one of the two solids.