Lesson video

Hello, my name is Mr. Chan.

And in this lesson, we're going to look at volume and further problem solving with spheres, cones, and pyramids.

Let's begin with a recap of the formula we need to know to work out the volume of the shapes we've got.

So we've got a cone, a rectangular-based pyramid, and a sphere.

So the volume of a cone, we work that out by the formula V equals one third, pi, r squared, h.

A rectangular-based pyramid, the volume is worked out by using one third, length, multiplied by width, multiplied by the perpendicular height.

And the sphere, we work out its volume using the formula four thirds, pi, r cubed.

Here's a question for you to have a go at.

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

Here's the answer.

So in this question, you need to work out how much empty space there is when the tennis ball is put inside the cube.

So, what you need to do is work out the volume of the cube, subtract the volume of the tennis ball.

The tennis ball is just a sphere, and that will be your answer.

Here's another question for you to try.

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

Here's the answer.

The tricky part with this question is working out the perpendicular height of the pyramid, and you can use Pythagoras to help you with that.

So think about where your right angle triangles are and how you can use Pythagoras to work out the perpendicular height in order to figure out what the volume of the pyramid is.

Let's have a look at this problem that involves the volume of a sphere.

So this example tells me that I've got 24 identical spheres, which have a combined volume of 4000 pi centimetres cubed.

And I know the volume formula for a sphere is four thirds, pi, r cubed.

Many of these problems involve trying to set up an equation to find out what we're actually asked to find out.

So, we're working out the diameter of each sphere.

So, I'm going to try and put the information into an equation.

The 24 identical spheres, so I've got 24 lots of the volume equals 4000 pi.

So I've said 24 multiplied by the formula for the volume of one sphere equals 4000 pi.

Now, I'm going to try and simplify this equation slightly by multiplying the 24 by four thirds to get 32.

So, 32, pi, r cubed equals 4000 pi.

I can see pi appears on both sides of the equation.

So, I'm going to divide both sides by pi.

That removes pi on both sides of the equation and I'm left with 32, r cubed equals 4000.

So divide both sides by 32.

We're left with r cubed equals 125.

And the inverse operation to cubing would be cube rooting, so I would cube root the 125 to find r equals five.

So, now I found the radius of the sphere.

It does ask me for the diameter.

So that would just tell me that the diamond is 10 and I'm doubling the radius to find the diameter.

Here's a question for you to try.

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

Here's the answer.

I hope you realised that to work out the volume of the hollow sphere, you need to work out volume of the larger sphere and subtract the volume of the smaller sphere.

And don't forget, you need to be working with the radius of these spheres.

Here's another question for you to try.

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

Here's the answer.

I think this question will need you to create some kind of equation to figure out what the radius of each cone is.

There is an example, very similar to this, earlier in the lesson, which you can have a look at if you're struggling.

That's all for this lesson.

Thanks for watching.