Video Transcript
In this video, we’re going to look
at how to calculate the area of a circle. So when we’re looking at the area
of a circle, we’re looking at the amount of two dimensional space within the circle
itself. Now there are two measurements that
we need to be aware of when we’re looking at circles. The first is the diameter of a
circle, so this is a line that starts at one point on the circumference, the edge of
the circle, and travels to the opposite side passing through the centre of the
circle. So for example, a line such as the
one I’ve drawn here, and this would be represented using the letter 𝑑 for
diameter. The other measurement we need to be
aware of is the line starting on the outside edge of the circle and just reaching
the centre of the circle.
So lines such as the one I’ve drawn
here, this is the radius of the circle and so would be represented using the letter
𝑟. So we want to look at calculating
the area. And there is a formula that we can
use in order to do that and it’s this formula here, which tells us that the area of
the circle is equal to 𝜋 multiplied by 𝑟 squared where 𝑟 represents the radius as
we said. Now this number 𝜋 is a very
special number in mathematics because of its relationship with circles. It’s called an irrational number
which means that its decimal representation has an infinite string of digits with no
repeating pattern. So if I was to try and write it out
as a decimal, I will be here forever. But it’s good enough to know that
𝜋 is approximately equal to 3.14. And so sometimes you’ll be asked to
use this value 3.14 as an approximation rather than the full value of 𝜋.
Now looking at the formula, it’s
really important to note the only the radius is squared. It’s not 𝜋 multiplied by the
radius and then square the result. It’s square the radius and then
multiply it by 𝜋. And you can see that if you think
about the order of operations, indices come before multiplication. So that’s our formula. Now we’ll see how we can use it for
a couple of questions. So here we have a circle and we’re
looking to calculate its area. Now we can see the radius of the
circle has been marked on for us; it’s 5.2 centimetres, so we just need to recall
our formula for the area. So that formula was area was equal
to 𝜋𝑟 squared. We just need to substitute the
value of 5.2 into this area formula.
So we have area was equal to 𝜋
multiplied by 5.2 squared. Remember, we’re just squaring the
5.2, not the 𝜋. And this will give me a value of
27.04𝜋 initially. Now this is what’s referred to as
giving your answer in terms of 𝜋 or as a multiple of 𝜋, which is often useful to
do if we haven’t actually got a calculator. But I’m gonna go one step further
and evaluate that, so multiply 27.04 by 𝜋. And that gives me a value of 84.9
rounded to one decimal place. Now notice the units are
centimetres squared because it’s an area that we were calculating. Okay, let’s look at a second
question.
So we’ve gotta a different
circle. Again we’d like to calculate the
area.
Now if you look carefully, we
haven’t been given the radius this time. We’ve been given the diameter, the
distance all the way across the circle. Now remember, our formula for the
area of a circle uses the radius. Its area equals 𝜋𝑟 squared. So we just have to remember that if
the diameter of the circle is equal to five, then the radius will be half of
that. They’ll be 2.5, because the radius
is always half of the diameter. So we have 𝑟 is equal to 2.5
centimetres.
Now we can just substitute that
value into the area formula. So we have area was equal to 𝜋
multiplied by 2.5 squared. And if I want to give my answer as
a multiple of 𝜋, it’s 25𝜋 over four, or I can evaluate that as a decimal. And it would give me a value of
19.6 centimetres squared to one decimal place. So when finally we have a circle,
just be very clear when you first look at the question; have you been given the
diameter or have you been given the radius? And remember that it’s the radius
that you need in order to use the area formula. Now let’s look at another type of
question.
The question says the area of a
circle is 28.3 centimetre squared. Find the diameter of the circle
to the nearest centimetre.
So this is an example of a
question where you’re working backwards from knowing the area to working out
either the radius or the diameter. So let’s just drop down our
area formula to start us off. Remember, the area is equal to
𝜋𝑟 squared and we’re told in the question that that value is 28.3. This means I can write down an
equation using the area of the circle that we’re given and the area formula. So here’s my equation, 𝜋𝑟
squared is equal to 28.3. Now what I need to do is solve
this equation in order to find the value of 𝑟. Now the question doesn’t ask me
for 𝑟; it asks me for the diameter, but remember they are very closely
related. So if I’ve got the radius, I
can find the diameter by doubling it.
So the first thing I need to do
is I need to divide both sides of this equation by 𝜋. And that will give me 𝑟
squared is equal to 28.3 over 𝜋. The next step, I’ve got 𝑟
squared and I’d like to know 𝑟, so I need to take the square root in order to
get 𝑟. So square rooting both sides of
this equation, I now have 𝑟 is equal to the square root of 28.3 over 𝜋, and at
this point I need my calculator to evaluate this value. Make sure when you’re typing
this into your calculator that you are doing the square root of this whole
fraction here, not just the square root of 28.3 and then dividing the result by
𝜋. It must be the square root of
that whole fraction. You could use brackets on your
calculator or the fraction button in order to make sure you were doing that
correctly.
So when I evaluate that I get
3.00136 and so on for the radius. Remember, I was asked for the
diameter, so I need to double this in order to work that out. So doubling that gives me
6.0027, and the question asked for the value to the nearest centimetre. Therefore, my answer is that
the diameter of the circle is six centimetre to the nearest centimetre.
So you can work both ways. You can calculate the area of a
circle from its radius or diameter or you can work backwards from knowing the area
to calculate the radius or diameter. In this case, it’s just a question
of forming an equation using the information that you’re given. Right. Now let’s look at a worded
question.
This question says a storm is
expected to hit seven miles in every direction from a small town. Giving your answer in terms of
𝜋, calculate the total area that the storm will affect.
So one thing that’s important
to note in this question is it asks us to give our answer in terms of 𝜋, which
means that our final answer shouldn’t be as a decimal but it should have 𝜋
involved in it. So often helpful just to do a
very quick diagram. So here is the town and we’re
told the storm is going to hit seven miles in every direction. So it’s forming a circle with a
radius of seven around this particular town. So here’s our area formula, the
area is equal to 𝜋𝑟 squared. So it’s going to be 𝜋
multiplied by, well in this case, seven squared. So there we have it, area is 𝜋
times seven squared. Seven squared is 49, so the
area is equal to 49𝜋. And that’s where I’m going to
stop because the question asks us to give our answer in terms of 𝜋.
So this would perhaps be the
type of question that you could answer without a calculator because you don’t
need a calculator to work out seven squared. And as you’re not actually
multiplying 49 by 𝜋, you could leave you answer like that. We do need some units, so the
units in the question were miles. Therefore, our answer is going
to be square miles for this area. So our answer in terms of 𝜋 is
49𝜋 square miles for the area that will be affected by the storm.
Okay, the next question.
A pendent is made out of
silver. Calculate the area of the face
of the pendant. And we have a diagram here
where the pendant is this shaded portion here.
So this pendant is a large
circle and then it has a smaller circle cut out to the middle of it. And it’s that sort of doughnut
shape that’s left that we’re looking to calculate the area of. So we can do this by working
out the area of the larger circle and then subtracting the area of the smaller
circle. Now the larger circle, if we
just think about it, has this radius here and that radius, well, it’s two
centimetres plus one centimetre. So this larger circle has a
radius of three centimetres. So our area formula is 𝜋𝑟
squared. And we’re gonna be doing the
large area subtract the small area. So we just need to substitute
in the relevant radii.
So for the larger circle, it’s
𝜋 multiplied by three squared and for the smaller circle it’s 𝜋 multiplied by
two squared. Now if we evaluate each of
these, we have nine 𝜋 subtract four 𝜋. And nine 𝜋 subtract four 𝜋 is
equal to five 𝜋. I haven’t been asked to leave
my answer in terms of 𝜋, so I’ll evaluate it as a decimal. And using my calculator, this
gives an answer of 15.7 centimetres squared to one decimal place. So that gives us the area of
this pendant here.
Okay, the final question we’re
gonna look at in this video.
The diagram shows a square of
side 12 centimetres with a semicircle added to one side and a quarter circle
added to another. We are asked to calculate the
total area of this shape.
So it’s a square of side 12
centimetres, so let’s just add that information onto the diagram. There we are. And we need the total area, so
we’ve got three areas to calculate, a square, a semicircle, and a quarter
circle. Now the square is easy enough,
so let’s do that first. To find the area of the square,
we just need to do 12 multiplied by 12. So we have 144 centimetres
squared for that part there. Now let’s think about this
semicircle. So we’re going to need our area
formula, which tells us the area is equal to 𝜋𝑟 squared. So let’s just think about the
radius of this circle. Well the total length of the
side of the square is 12 centimetres. The radius of the circle,
therefore this part here, must be equal to six centimetres.
So to work out the area of this
semicircle, we can work out the area of a full circle of radius six, but then we
need to divide it by two as we only have half of that circle. So we have the area of the
semicircle is equal to 𝜋 multiplied by six squared divided by two. And if I work that out as a
multiple of 𝜋 initially, it will give me 18𝜋 as the area of that
semi-circle. Okay let’s turn our attention
into the quarter circle. So to work out the area of the
quarter circle, we can find a full circle and then divide it by four. Now the radius of the quarter
circle, so this part here, is the same as the side length of the square. So the radius this time is 12
centimetres. So to work out the area, we’ll
do 𝜋 multiplied by 12 squared, and then we need to divide it by four as it’s
just a quarter circle. Again, I work this out just in
terms of 𝜋 initially. So this gives me an area of
36𝜋 for that quarter circle.
The final step then is to work
out the total area by adding these three individual areas together. So we have 144 plus 18𝜋 plus
36𝜋, which gives me an answer of 144 plus 54𝜋 centimetres squared, if I was
going to leave my answer in terms of 𝜋. But I’ll evaluate it as a
decimal, which gives me a final answer of 313.6 centimetres squared and that’s
to one decimal place.
So we’ve looked at calculating the
area of the circle from the radius or diameter. We’ve looked at working backwards
from knowing the area to calculating the radius or diameter and then a couple of
problems associated with areas of circles.
