Lesson video

Hello, I'm Mrs. Lashley and I'm gonna be working with you as we go through the lesson today.

I really hope you're looking forward to it and you're ready to try your best.

By the end of the lesson, hopefully you can calculate the surface area of a sphere.

We have a new keyword today which is a sphere.

So a sphere is a 3D shape where every point on its surface is equidistant from the centre and it's in the lesson title.

It's in our outcome, we'll be using it a lot during the lesson.

However, we are also looking for the surface area.

And surface area is something you have calculated previously for other 3D shapes.

So you may wish to pause the video and just reread the definition of surface area before we make a start.

So in our lesson for finding the surface area of a sphere, we're gonna break it into two learning cycles.

The first learning cycle is to be able to find the surface area of a sphere.

And in the second learning cycle we're gonna use the surface area of the sphere.

So let's make a start working out the surface area of a sphere.

So as we've already said, a sphere is a 3D solid where every point on its surface is equidistant from its centre.

Can you think of any everyday objects that can be described as being the shape of a sphere or maybe you say spherical.

So pause the video, you might wanna have a look around the room you're in or around your house to find some spherical objects.

Press play when you're ready to move on.

So you may have thought of a bowling ball, a grapefruit, the planet earth.

Technically the planet earth is not a perfect sphere, but we probably would describe it as spherical and we definitely would model it to be a sphere.

Lucas has heard that the surface error of a sphere is equal to four Pi R squared.

So it's gonna try and see if it is true by using an orange.

So Lucas says I will model the orange as a perfect sphere.

The orange pill covers the surface, so I'll use this to see if it's equivalent to four Pi R squared and R is that radius.

So remember that a sphere, a perfect sphere is where all the points on the surface are a fixed distance from its centre.

And we would call that a radius in a similar way to we'd call it a radius on a circle.

The 2D shape.

So four Pi R squared equals four times Pi R squared.

We can think about that term as a product of four and Pi R squared.

And Pi R squared is the area of a circle with the same radius as the orange.

So Lucas is going to draw four circles because it's four times Pi R squared, which are the same size as the plan of the orange.

So it is gonna draw around it four times.

So that's the start.

Lucas is trying to show that the surface area of this orange, which is the peal, is equivalent to four Pi R squared, where R is the radius of our orange and the radius of these circles.

So I'm gonna use the orange peel, Lucas says, which is the surface area 'cause it covers the surface of the sphere, to see how many circles it can fill.

It should fill all four.

So he's peeled his orange, breaking into smaller bits and now distributing it across the four circles.

So he's filled one, he's filled two, three, four.

So Lucas says obviously it isn't perfect, obviously they're not perfect circles, but this isn't a proof, but it does seem to confirm that the surface area of a sphere is calculated by doing four Pi R squared.

So this is a demonstration of the formula four Pi R squared for the surface area of a sphere.

So here's a check.

Which of the following calculates the surface area of a sphere with a radius of R? Is it A, B, or C? So pause the video and when you're ready to check, press play.

It's A, so four Pi R squared, you can think of it as four areas of a circle.

And the area of a circle is Pi R squared, the R, the radius is what is being squared.

So now I'm gonna go through an example of calculating the surface area of a sphere and then there'll be one for you to try.

Calculate the surface area of a sphere with a radius of six centimetres and there's a little diagram there to indicate a sphere with a radius of six.

So the surface area is four times Pi R squared, R is six, that's our radius.

So four times Pi times six squared, six squared.

We can evaluate indices first, 36, and then we can do our product, four times 36, 144 times by Pi.

And so this is the exact surface area, leaving it in terms of Pi, this would be, if we didn't have a calculator, we would leave it in terms of Pi, 144 Pi square centimetres.

If you could use a calculator or it asks you to give it to a particular degree of accuracy, then 452.

4 to one decimal place is the surface area of this sphere.

So here's one for you to try.

So pause the video and then when you're ready to check, press pay.

So your surface area is gonna be calculated by doing four times Pi times eight squared because the radius is eight.

Evaluating that gives you 64, four times 64 is 256.

So in terms of Pi, 256 Pi square centimetres or to one decimal place, 804.

2.

We're gonna go through the same thing again, I'm gonna go through one and then one for you to try, but this time it's slightly different.

Calculate the surface area of a sphere with a diameter of 10 centimetres.

So the diameter is going from the surface to its direct opposite point on the surface, through the centre.

So it's going to be the widest point of the sphere.

So the diameter is 10 centimetres, so the radius would be five.

In a similar way to how we use diameter and radius for a circle.

Surface area for a sphere is four Pi R squared.

So we do need the radius.

So we can now substitute that into the formula, four times Pi times five squared, evaluate five squared gives us 25, four times 25 is 100.

So in terms of pi, 100 Pi square centimetres, that is the surface area of this particular sphere to one decimal place that's 314.

2 square centimetres.

Here's one for you to do.

Pause the video and then when you're ready to check, press play.

So once again, you have the diameter given to you, so you needed to go from the diameter to the radius by halving it.

Now you can use the formula and substitute it in.

So 400 Pi was your exact answer or 1256.

6 square centimetres to one decimal place.

If you've rounded to a different degree of accuracy, just check it against the one decimal place.

So up to the first task of the lesson for you.

And so question one is to match the diagram to the correct calculation for the surface area of the sphere.

So there's four diagrams, A, B, C, and D, and there are four calculations.

The calculations have not been evaluated, they are sort of midway work, so you need to match them up, pause the video and when you're ready to move on, press play.

Question two then, so find the surface area of the following spheres, leaving your answers in terms of Pi.

So A, B, C, and D have all been given as a diagram, E and F have been given to you in words.

So pause the video, work through and get those surface areas.

In terms of Pi, you should be doing this without a calculator.

And then when you're ready for the final question of the task, press play.

So the final question, question three.

By modelling planets as spheres, complete the table.

So the planets are down the left hand column, then we've got the radius in kilometres, the diameter in kilometres, the surface area in terms of Pi and the surface area to three significant figures, you will be working with standard form because of how large the values become.

So pause the video, you can use a calculator for this one.

And when you're ready to go through the answers to task A, press play.

Question one, you were matching up.

So diagram A matched with the calculation E.

That one was written in this way of the formula.

So four times Pi times radius squared.

B matched with F, 18 times four times Pi times 18.

So this was not written as four times Pi R squared.

R squared means R times R.

So we can see 18 times 18 within the calculation.

So that's the 18 squared.

And then four times Pi, which was the constant in the formula.

C matches with H.

C has a diameter given to you.

So we needed to go, okay, the diameter is 10, the radius is there for five.

And if we look at H, the calculation is written as four times Pi times radius, times radius and radius times radius is radius squared.

Which process of elimination means that D and G go together.

G says Pi times two squared times four.

So there's just a reorder of the product there.

The diameter was four, which meant the radius was two.

Two squared gives us the radius squared part of the formula and the surface area.

Question two, you needed to find the surface area for the following spheres, leaving your answers in terms of Pi.

So for A, the radius was three, it was just a case of substituting it in and evaluating.

So three squared is nine, nine times four is 36.

36 Pi, for B, seven squared is 49, 49 times four, 196 Pi square units, for C, noting that it was the diameter that was 12.

So halfing it to get the radius before evaluating and finding 144 Pi and D, the diameter again, so half it gives you nine, nine squared times by four, 324 Pi square units.

E and F didn't have a diagram.

You could have sketched yourself a diagram if you find that useful.

E was a radius of a half.

Some of you may have thought, well I can't do this with a calculator.

Well 0.

5 is the decimal equivalent of a half.

If you think of a half as the fraction a half, when you square it, you get one quarter, a half times a half.

So a quarter times four is one.

And then so one Pi, Pi is the answer and F, the diameter was three.

So you needed to half it to get the radius, which is 1.

5.

Again, you could write that as an improper fraction.

Three halves, when you square that you get nine quarters.

Nine quarters times four is nine.

So nine Pi is the surface area in terms of Pi.

So sometimes using the decimal is absolutely fine, but a fraction might be an easier way to do it without a calculator.

Question three, I would encourage you to pause the video and just check your cell by cell, use your rows for the mercury, for Venus, et cetera.

So to move from the radius to the diameter, you can clearly half or double depending on which way you're going.

The surface area is four times Pi R squared, and both of those is the same, but one was in terms of Pi and one was rounded.

I rounded to three significant figures.

So as I say, pause the video, check it carefully before you press play to move on.

So our second learning cycle is making use of the surface area of a sphere.

So a sphere has a surface area of 576 Pi square centimetres.

Jun says, there's only one sphere with that surface area, it would have a radius of 12 centimetres.

Alex says, how do you know that? So can you think how Jun knows that the sphere has a radius of 12 centimetres? Jun goes on to say, well there is only one variable that influences the surface area, which is the radius, four times Pi times radius squared.

The four and the Pi are constants, the radius is the variable and the surface area depends on that variable.

Alex says, okay but how do you get to 12? Have you figured out how we get to 12 yet? So Jun has set up an equation.

The surface area of any sphere is four times Pi times radius squared, and this is equal to 576 Pi in this particular sphere, he's divided both sides by four using inverse operations to solve the equation.

So now Pi times radius squared equals 144 Pi divided by Pi on both sides to keep it balanced and therefore the radius squared is 144 and we can square root and get the positive square root as 12 because this is a length.

Alex says, ah yes, that makes sense.

So if we have the surface area for a sphere, then we can work to this radius.

So here's a check for you.

A sphere has a surface area of 100 Pi.

What is the radius of the sphere? Pauses the video, be careful.

And when you're ready to check, press play.

So the radius of the sphere is five.

I do wonder how many of you went for 10? And that's because you may have noticed 100 as a square number and just square rooted.

But remember the surface area is four times Pi R squared.

So we did need to divide by four to get Pi R squared, which was 25 Pi.

And then we can recognise that R squared would be 25 and the positive square root would be five.

So do be careful.

Remember we need four Pi R squared for the surface area.

So if we can work out the radius from knowing a sphere's surface area, what else can we work out? Have a moment to think about that.

So if you can go from surface area to the radius, what else can you work out? Well, you can also work out the diameter of the sphere as the diameter is twice the radius.

So if we just look at one of those, a sphere has a surface area of 196 Pi square centimetres.

What is the diameter of the sphere? What we can go through the same steps to get to seven.

So dividing by four, dividing by Pi, square rooting.

But the question wants the diameter.

So if the radius is seven then that means the diameter is 14 and the units would be centimetres.

Check for you, a sphere has a surface area of 1,444 Pi square centimetres.

What is the diameter of this sphere? Again, pause the video and when you're ready to check, press, play.

So we are going to use the surface area and the formula to get to the radius.

The radius is 19 and then if we know the radius, we can double it to get the diameter.

So hopefully you got the diameter was 38 centimetres.

So the earth is one of the planets, our planet, what we live on, and it can be described as having two hemispheres, the northern and southern hemisphere.

So here's the north and here's the south, it's the equator that sort of breaks that, or it could be eastern and western hemispheres.

So what is a hemisphere? We use that in our sort of geographical language, but what is a hemisphere? Well Hemi, the prefix is Greek for half.

So a hemisphere means half sphere or half of a sphere.

So we can see here the northern and southern if you like, of our globe, two hemispheres.

So if a sphere has a surface area of 1,296 Pi square centimetres, what would be the curved surface area of a hemisphere with the same radius? Well the hemisphere is half the sphere, so the curved surface area will be half of the surface area of the sphere.

So in this case, it would be 648 Pi.

We just divide the surface area by two, we half it remembering that hemi is Greek for half.

A check for you.

A sphere has a surface area of 676 Pi square centimetres.

What is the curved surface area of a hemisphere with the same radius? Pause the video.

When you're ready to check, press play.

The curved surface area would be the same radius, would be 338 Pi square centimetres.

So you just need to divide it by two.

So if a sphere has a surface area of 484 Pi square centimetres, what is the total surface area of a hemisphere with the same radius? So the wording has slightly changed.

Previously I was saying curved surface area.

Now I'm saying total.

So what does that mean? Well the total surface area is both the curved surface area, so the sort of dome part, the bit that would be the surface of a sphere, as well as the circular surface area.

So the circle at the base, the curved surface area of this hemisphere would be 242 Pi, as it's half the surface area of the sphere.

And the circular face can be calculated to have a radius of 11 centimetres.

How would we get that 11 centimetres? Well we know how to get from the surface area to a radius.

Remember we were divided by four, divided by Pi and then square root it and the radius of the sphere will be the same radius as the circular base of the hemisphere.

So it's got a radius of 11 centimetres, which means that the circular face has an area of 121 Pi.

And so our total surface area is the curved surface area of the hemisphere and that circular face so we can add them together.

So here's a check for you.

A sphere has a radius of 15 centimetres.

What is the surface area of this sphere? Pause the video When you're ready to move on, press play.

So four times Pi times radius squared.

We're back to full spheres.

We're not looking at hemisphere in this question.

A full sphere.

So four times Pi times radius squared is the surface area formula and that evaluates to 900 Pi square centimetres.

Hence what is the curved surface area of a hemisphere with a radius of 15 centimetres.

Pause the video and when you're ready to move on, press play.

Well be half of that because hemisphere is half of a sphere and we're just focusing on the curved surface area.

So 450 Pi square centimetres.

The total surface area of a hemisphere with a radius of 15 centimetres is how many Pi? So pause the video and think about what else you need to get that total.

Press play when you're ready to check it.

Be 675 Pi.

Why, because you need that circular base for the total surface area.

The radius is 15, so Pi times 15 squared and then add that onto the curved surface area.

So 675 Pi comes from 450 Pi plus 225 Pi.

So we're onto the task now, question one, two and three are all on the screen here.

So question one, a sphere has a surface area of 2,116 Pi.

Part A is what's the radius of the sphere and part B is hence what's the diameter.

Question two, a sphere has a surface area of 1,156 Pi square centimetres.

What is the diameter of the sphere? And question three, a sphere has a surface area of Pi square metres.

What is the curved surface area of a hemisphere with the same radius? Pause the video and then when you're ready to move on with the questions, press play.

Question four, five and six are now on the screen.

So question four has got three parts A, B and C.

So work through all of those, question five radius of the hemisphere, given that the curved surface area is 338 Pi square centimetres.

And question six, a hemisphere has a radius of R centimetres.

What is the total surface area of the hemisphere? Pause the video and then when you're ready for the answers to task B, press play.

So question one had two parts.

A sphere has a surface area of 2,116 Pi square centimetres, what is the radius of the sphere? So set up our equation, we got the formula four times Pi R squared and equal to the given surface area.

We can then solve that to get that the radius of the sphere is 23 centimetres.

Part B was hence what is the diameter of the sphere? Well if the radius is 23, the diameter is double that, 46 centimetres.

Question two, a sphere has a surface area of 1,156 Pi square centimetres.

What is the diameter of the sphere? So you need to go through the same steps as question one.

Set up the equation, solve it, radius is 17.

So if the radius is 17, the diameter would be 34 centimetres because remember, the diameter is twice the length of the radius.

Question three, a sphere has a surface area of Pi square metres.

What is the curved surface area of a hemisphere with the same radius? Well the hemisphere is half the sphere, it's the curved surface area.

So we are going to half it and it would be half Pi square metres, 0.

5 Pi, or you could have written that as Pi over two or you could have written that as a half Pi using a fraction.

Question four has got three parts.

A sphere has a surface area of 2,704 Pi square centimetres.

Part A, what is the radius of the sphere? So we've done that a couple of times already within this task.

So I'm hoping you're being successful.

The radius is 26.

Part B, what is the curved surface area of a hemisphere with the same radius? So we're going to half the surface area of the sphere, which is 1,352 Pi square centimetres.

And then part C is what is the total surface area of the hemisphere with the same radius.

So the circular face of the hemisphere will be 676 Pi because that will be 26 squared times Pi.

And the total surface area is therefore the curve surface area which you worked out in part B plus that circular face.

So that's 2028 Pi square centimetres.

Question five, a hemisphere has a curved surface area of 338 Pi square centimetres.

What is the radius of the hemisphere? So this one we haven't done any questions but you've got the right skills to do it.

A sphere with the same radius would have a surface area of double the hemisphere, which is 676 Pi square centimetres.

And we know we can find a radius if we have that surface area of a sphere.

So four times Pi times radius squared equals 676 Pi and then we can solve it to get the radius is 13 and the radius of the hemisphere and the sphere would both be 13 centimetres.

Lastly, question six, a hemisphere has a radius of R centimetres.

What is the total surface area of the hemisphere? So the surface area of the sphere with radius R is four Pi R squared, square centimetres.

The curved surface area of a hemisphere with the same radius would be half of that, which would be two Pi R squared centimetre squared.

The area of the circular face on the bottom of the hemisphere would be Pi times radius squared 'cause it's a circle and the radius is R, so Pi R squared.

And the total surface area would therefore be that circular face plus the curved surface area, two Pi R squared, which comes to a total of three Pi R squared square centimetres.

And perhaps you noticed as you were working out previous questions, the total surface area of the hemisphere is 3/4 of the sphere's surface area.

So to summarise today's lesson, which was working out the surface area of a sphere, the surface area of a sphere can be calculated using a formula where it's four Pi R squared, and R is the radius of the sphere.

A hemisphere is half of a sphere.

The total surface area of a hemisphere is the curved surface area, plus the area of the circular face.

Really well done today and I look forward to working with you again in the future.