Lesson video

Hello and welcome to this lesson on revisiting area, and today we'll be talking about the area of circles.

Now before you start this lesson, make sure you have a pen or a pencil, and something to write on.

Okay, now that you have all those things, let's get on with today's lesson.

Firstly, try this activity.

Now these are circles drawn on a unit grid.

Now what that means is that each square is worth one unit.

In this instance the units are centimetres.

So by counting squares, can you state an approximate area for each of these circles and then what do you notice about that approximate area and the radius? Pause the video here and give this a go.

Okay, now that you've tried this, let's see what you've come up with.

Now for that first circle when you counted the squares, you should have gotten approximately 12 full squares.

Now you may have gotten just over 12.

But you should have approximately counted 12 four squares.

For that second one, you should have approximately counted 27 full squares.

And then for that last one, you should have gotten around.

Can you see any relationship between the approximate number of full squares and the radius of this circles? Well, if you look at this, in that first one the number of full square is what? Six times the radius, and the second one, the number of full squares is nine times the radius.

And then in that final one, the number of full squares is.

Now right now nothing seems really obvious as jumping out as us, 'cause the first one is six, the second one is nine, the third one is 15.

What do six, nine and 15 all have in common.

Good, they're all multiples of three.

So that first one that is two times three, that second one is three times three, and that last one is five times three.

Are you I you spotting something? So if we look at this, the number of the approximate number of squares that we've counted.

What is the radius? This instance radius was two times the radius again times three.

And the second one is the radius, times the radius again, or times three.

The last one is the radius, times the radius again times three.

So the approximate number of squares.

Now we going to to look at how we actually work out the actual area as we go on in today's lesson.

Now it turns out to work out the area of a circle.

The formula is pi times the radius squared.

Now this number pi is approximately 3.

142, and this is to three decimal places.

Now pi is the ratio between the diameter of the circle.

So it's the ratio between the diameter of the circle and the circumference of the circle.

So it's a number that's constant for all circles.

It's approximately 3.

142.

That's why we noticed here that the number of four squares was three times bigger, approximately about three times bigger than the radius squared.

So in order to calculate the area, you always do pi times the radius squared.

For a circle with a radius of eight centimetres.

What is this area? Well, it would be pi times eight squared.

pi times a squared eight squared.

Eight squared is 64.

So this will be written as 64 pi centimetres squared.

Now we leave our answer here in terms of pi, because it's the most accurate answer.

Now, here we are.

Here we have another circle drawn on a new grid and a triangle also drawn on a unit grid.

So which of these two has a bigger area? What do you think? Well, just by looking at it you may say, "Well, the triangle looks like it covers more area." Looks like it's covering more space.

For what is the area of that triangle? Well, if you look the base is three and the height is one, two, three, four, five, six, seven, eight, and the height is eight.

So the area of that triangle is three times eight divided by two which is 12 centimetres squared.

We know centimetre squared 'cause we're on a centimetre grid.

So each square is worth one centimetre.

Now, would the circle with the radius of two have a smaller or larger area? Well, remember before we said the approximate number of squares in there was 12 squares, right? So this was 12 unit squares, should approximate area was 12 centimetres squared.

But how do we know the circle has a bigger area? Good, I think that's the formula.

The formula for the area for circle is pi times the radius squared.

Now radius squared is four.

So the area of this circle, the precise area of this circle pi times four, which is four pi centimetres squared.

Now how do we know four pi is bigger than 12? Well, 12 is four times three, so 12 is four times three.

And we know that pi is just a little bit bigger than three.

So that's how we know four pi is bigger.

Now here's an independent task, for these circles give you areas in terms of pi.

And then on the second question, draw the largest triangle with integer side lengths, that has an area smaller than that of a circle with a radius of three.

Pause the video here and give that a go.

Okay, now that you've tried this, let's see what You've come up with, well, a circle with the radius of four centimetres.

Well, that is pi times four squared, that is 16 pi centimetres squared.

A circle the radius of eight.

Well, that is pi times eight squared which is 64 pi centimetres squared.

Now we want a triangle with integer side lengths with an area smaller than a circle with the radius of three.

Well, what's that area of a circle the radius of three? hat's pi times three squared which is nine pi centimetres squared.

Well, nine pi is bigger than 27.

Because pi is just a little bit bigger than three.

So nine times pi is bigger than mine times three.

So we want a triangle with an area of 27.

So what are the five lines of a triangle? So we're using a unit grid, you can just draw a triangle that has 27 squares in it, or you can try and work it out.

Well, if the area is going to be 27, and we know that say one of the sides is nine, so one, two, three, four, five, six, seven, eight, nine.

So nine times something divided by two is 27.

Well, if you double 27 you get 54.

So what nine times something gives you 54, so that's nine times.

Good, that's nine time six.

And if you join a nine by six triangle, you count the squares in there.

It is 27 full squares.

If you would worked out the area, it's nine times six divided by two, that is 27 centimetres squared.

Now that is the largest triangle with integer side lengths, so one side is six centimetres and the other side is nine centimetres.

Okay, now has an explore task.

So you going to draw three different shapes with an area bigger than a circle with a radius of four centimetres.

This has to be three different shapes.

Pause the video here and give this a go.

Okay, now that you've tried this, let's see what you come With.

Well, the area of a circle with the radius of four, that is pi times four squared, which is what? 16 pi.

And 16 pi, remember is about 16 times 3.

142 approximately.

So you want areas, that are constantly bigger than 48 to be sure, that you have an area that's bigger than the circle, because if you do an area of 49 more, you will notice if you actually calculate this, an area 49 is less than 16 pi 'cause 16 pi is about 50, rounded to the nearest whole number.

So you want areas that are comfortably bigger than 48, So let's do one with an area of 60.

So you can do a rectangle with an area of 60, so that is six by 10 rectangle, that has an area 60 and you know that that's bigger than 16 pi.

Well, if we know that an area of 60 becomes 60 pi, we can draw a triangle with an area of 60.

I'm not going to draw the triangle to scale we don't have enough space on the grid.

So we can say not drawn to scale.

So the hieght of 10 and a base of 12, and that triangle has an area of 60 centimetres squared.

Again, we're looking for another shape with an area of 60 centimetres squared.

So what else can we do? Well, we can do a parallelogram with a base of 10 and a height of six.

one, two, three, four, five, six, and a height of six.

Again, this is just going to go off the page just a little.

There we go, and those are all three shapes with an area of 60 centimetres squared.

And we know that 60 centimetres squared is comfortably bigger than 48.

So we can definitely be sure that all of these are bigger than 16 pi.

That's if we don't have a calculator to calculate how much 16 pi is exactly.

Okay, now I really want that for getting through this lesson.

And I look forward to seeing any work that you have to share and what shapes you came up with that were bigger than 16 pi.

So if you want to share your work, ask your parent or carer to share welcome Twitter, tagging @OakNational and #LearnwithOak.

I will see you again next time.