Lesson video

Hello, my name is Mr. Chan.

And in this lesson, we're going to learn about the surface area and volume of a hemisphere.

Let's begin with something that we should already know.

So if we want to work out the volume of a sphere, that is given by the formula, V equals 4/3rds pi r cubed.

So that's the volume of a sphere with radius r.

Now for a hemisphere, a hemisphere is simply, just half of a sphere.

So imagine cutting the sphere in half.

That's what the hemisphere is.

So if we're halving the sphere to get hemisphere then the volume similarly will be halved.

Now, we could find the volume of a sphere first and then half it, but if we have the volume of a sphere formula, this is what the volume of a hemisphere formula would look like, because really, we're only halving the fraction 4/3rds.

Halving the fraction 4/3rds gives us 2/3rds.

So this formula here will help us find the volume of a hemisphere.

So let's look at an example where we use this formula for volume of the hemisphere.

And here we've got an example.

to work out volume of the hemisphere, we've got to leave answer in terms of pi.

So the diagram shows us a hemisphere with a radius of six centimetres.

And, our volume formula for hemisphere is 2/3rds pi r cubed.

So the first step is just to substitute the radius into the formula.

So we get our next line working out to be 2/3rds multiplied by pi multiplied by six cubed.

Working out what six cubed is.

Our next line will help us to calculate the volume, to be 2/3rds multiplied by pi multiply by 216.

To give a final answer of 144 pi to get the 144.

I've just multiplied the 2/3rds and the 216 together.

And we're leaving pi as it is because we'll be leaving our answer in terms of pi.

So the volume of the hemisphere is 144 pi centimetres cubed.

Now if the question wanted us to give our answer to one decimal place, we could use our calculator to figure out, what 144 multiplied by pi is.

That will give us an answer of 452.

4 centimetres cubed, and that's rounded to one decimal place.

Here is a question for you to try.

Remember, we are leaving your answer in terms of pi in this question, so pause the video to complete the task, resume the video once you're finished.

Here's the answer.

In this question, you're trying to find the volume of a hemisphere with a radius of five centimetres.

If you use that to substitute that radius into the formula for the volume of hemisphere, you shouldn't have any problems. Now let's look at how we work out the surface area of a hemisphere.

So we use what we should already know about a sphere, and its surface area.

And that's given by the formula, the surface area of a sphere equals 4 pi r squared.

So applying that to a hemisphere, remember, us hemisphere is just half of a sphere.

So, really, we can just halve the formula for the surface area of a sphere to get the surface area of a hemisphere.

So halving that formula will give us 2 pi r squared.

As you can see, we've just really halved what we're multiplying by in terms of the constant value 4, so we get 2 pi r squared.

However, just remember, this is the curved surface of the hemisphere.

And we have to look at, the base of the hemisphere as well if we want something called the total surface area.

Let's look at an example where we try and work out the surface area of a hemisphere.

Now that we know how to work out the curved surface area of the hemisphere.

So in this example, again, we've got a hemisphere that's got raised six centimetres, and we're going to leave our answer in terms of pi in this question.

So the curved surface area is given by that formula, 2 pi r squared.

So we substitute the radius into the formula to get 2 pi, 2 multiplied by pi multiplied by 6 squared.

So figuring out what 6 squared is 36 and we can calculate 2 multiplied by 36.

And the final answer for the curved surface area is 72 pi centimetre squared.

Now, think about when you do cut a sphere in half, you end up with a circular face, which forms the base of the hemisphere, so you can see that shaded in there.

Now, that's just a circular face, which we can figure out the area of a circle by the formula pi r squared.

So again, we know what the radius is.

So let's substitute the radius into this formula for the circle area.

So we get pi multiplied by 6 squared, that gives us pi multiplied by 36.

And the area would be 36 pi said to be 2 squared.

So all that's left now, is to add those two together to give a total surface area of the hemisphere 72 pi, add 36 pi gives us a total surface area of 108 pi centimetre squared.

Here is an example free to try.

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

Here are the answers.

I hope you realised in this question that you are given the diameter of the hemisphere and not the radius.

So to get the radius, you're going to halve to have the diameter.

So you should be working with the radius of 13 millimetres in this question.

Here's another question for you to try.

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

Here are the answers.

Again, be careful in this question because you've been given the diameter of Jamie's bowl in the shape of a hemisphere.

So the 12 centimetre bowl actually has a radius of 6 centimetres.

In part B, you're going to have to work out volume to see whether you can fit, 480 centimetres cubed of soup into that bowl, and looking at the answer.

It looks like you can't, because the bowl is too small.

Here's another question for you to try.

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

Here's the answer.

This question tells you about an ice cream on top of a cone and the ice cream is in the shape of a hemisphere, and you need to work out the volume of ice cream that's on top of the cone.

So think about what you need to know in order to work out volume of a hemisphere, you need to work out the radius.

So it's told you also that the cone has a height of 11 centimetres and the total height of 15 centimetres.

How can you figure out what the radius of the hemisphere is, using the information there? That's all for this lesson.

Thanks for watching.