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 30 centimetres cubed and, in the
case of cuboid B, four times six times ten, which is 240 centimetres cubed. Now as we did with the lengths,
let’s write a ratio for these two volumes. So it will be 30 to 240, but that
simplifies because both sides of that ratio can be divided by 30. 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 343. 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
cones 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 70 centimetres cubed, and multiplying it by eight. And doing so, it gives me an answer
then of 560 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 16 to
54. And we can simplify that by
dividing both sides of the ratio by two. So the simplified ratio is eight to
27.
Now looking to work out the height
of the smaller pyramid, which is the 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 27. 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.
