Lesson video

Hello, my name is Mr. Chan.

And in this lesson, we're going to learn how to find volumes of similar shapes given corresponding lengths.

Let's have a look at this example first.

We've got cuboid A and cuboid B which are similar.

So I could see those cuboids there and we need to work out the length scale factor first.

So that's also known as the linear scale factor, so if you see some questions asking if the linear scale factor is the length scale factor.

So we can use corresponding sides for cuboid A and cuboid B to work this out.

So I can use the base, which is six centimetres and 12 centimetres, I can see, I multiply that by two to get from six to 12.

So that would be my scale factor two.

Now to work out the height of cuboid B I could use that scale factor for the length.

So looking at the height that corresponds with three centimetres in cuboid A.

So I'll multiply that by two, that gives me six centimetres as a height for cuboid B.

In order to work out the volume of each shape I would multiply the three lengths together so there's the length of the base multiplied by the height, multiplied by the width.

So we've got the six multiplied by two, multiplied by three, to give a volume of 36 centimetres cubed.

Do exactly the same for cuboid B.

So we multiply the base multiplied by the width, multiplied by the height to give a value of 288 centimetres cubed for the volume.

Now, if we look at the volume scale factor from A to B, so how many times bigger is volume B from A? So we can look at and compare these two values, and if we did a division calculation to work out what that is multiplied by eight, what do you notice? Here's another example.

We've got two cylinders, cylinder A and cylinder B which are similar.

So one's an enlargement of the other, and we can work out the length scale factor from A to B by looking at corresponding lengths.

So I've got the radius values to work with.

I've got four centimetre radius for A and 12 centimetre radius for B.

So I can work out the linear scale factor using these two lengths.

So I can see that four multiply by three to get me to 12 so that length scale factor is three.

Now I can use that to work out the height of cylinder B so I can see the height of cylinder A is five centimetres, multiply that by three, that will give me the height of cylinder B to be 15 centimetres.

To work out the volume of each shape, just to remind you that the volume of a cylinder is Pi r squared h.

So I've got the radius and I've also got the height of each cylinder.

So I can work out the volume of A Pi multiply by four squared, multiply by five, that gives me a volume of 80 Pi centimetres cube.

And I've left the answer in terms of Pi 'cause it's just more straightforward and more accurate.

So the volume of cylinder B would be Pi multiplied by 12 squared multiply by 15, that gives me a value of 2,160 Pi centimetres cubed.

Now, in order to answer this question, what is the volume scale factor from A to B? I can simply divide the volume of B divided by the volume of A to tell me how many times bigger it is.

What is the multiple? So the volume scale factor in this example is 27.

What do you notice? Here's a question for you to try.

Pause the video to have a go, resume the video once you're finished.

Here is the answer.

So once you found the scale factor for the length is two, you now know that the volume will be four times bigger.

So we're told the volume of prism P to be 80 centimetres cubed.

So the volume of prism R is four times bigger, which is 320 centimetres cubed.

Here's another question you can try.

Pause the video to have a go, resume the video once you're finished.

Here's the answer.

So when you found the linear scale factor is three, the volume scale factor is 27.

So let's try and sum up what we've learned so far.

In our cuboid example, we had two cuboids that were similar and the length scale factor was two, and we found the volume scale factor to be eight.

In our cylinder example, we found the length scale factor to be three and the volume scale factor to equal 27.

Now the shapes don't really matter in these examples, it's the values that link the length scale factor, and the volume factor together for three dimensional solids that's important.

What you will find is that the volume scale factor is the length scale factor multiplied by itself three times.

So if we look at the cuboid example, the length scale factor is two, so if we multiply that length scale factor by itself three times, two times two times two, that gives me the volume scale factor of eight.

In my cylinder example, the length scale factor is three, so to work out my volume scale factor, I multiply the length scale factor by itself three times, three multiply by three, multiply by three.

That gives me a volume scale factor of 27, and really all we're doing is cubing the length scale factor.

So let's put that into another example we've got here.

We've got two regular hexagonal prisms, which are similar.

So the length scale factor we can see would be five.

To work out the volume scale factor, we would cube the length scale factor.

So we're multiplying that by itself three times, five cubed, which equals 125.

What that tells me is that the larger prism there, prism B, would be 125 times bigger in terms of the volume.

So the volume, if we're told is 60 centimetres cubed in prism A, that means the volume of prism B will be 125 times that, so 60 multiplied by 125 gives me 7,500 centimetres cubed.

Here's an example you should try.

Pause the video to complete the task, resume the video once you're finished.

Here's the answer for question three.

In this question, you're told that the height of the vase has doubled.

So that means the length scale factor is two.

But when you're working out the volume scale factor, remember, you've got to cube the length scale factor to work out how many times bigger the volume is.

So that's why in this question, Alisha's wrong.

So the volume scale factor is two to the power of three, which is eight, so the volume of the vase that's larger is eight times bigger, it can hold eight times more water.

Here's another example for you to try.

Pause the video to complete the task, resume the video once you're finished.

Here's the answer.

This question is slightly different to the other questions, because you're told what the volumes are, and you have to work out what the volume scale factor is in this question first.

The volume scale factor in this question is 27, and that will then tell you that the linear scale factor is three.

You could work that out by cube rooting the 27, and then once you found that you can find the missing side length.

That's all we have time for this lesson, thanks for watching.