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Video: Areas of Circles

Lauren McNaughten

The definition of the area of a circle. Find the area of a circle given its radius or diameter. Find the diameter or radius of a circle given its area. Work to a specified number of decimal places or give answers in terms of pi.

13:31

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 three point one four. And so sometimes you’ll be asked to use this value three point one four 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 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 five point two 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 five point two into this area formula.

So we have area was equal to 𝜋 multiplied by five point two squared. Remember we’re just squaring the five point two, not the 𝜋. And this will give me a value of twenty-seven point zero four 𝜋 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 twenty-seven point zero four by 𝜋. And that gives me a value of eighty-four point nine 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 the 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 two point five, because the radius is always half of the diameter. So we have 𝑟 is equal to two point five centimetres. Now we can just substitute that value into the area formula.

So we have area was equal to 𝜋 multiplied by two point five squared. And if I want to give my answer as a multiple of 𝜋, it’s twenty-five 𝜋 over four, or I can evaluate that as a decimal. And it would give me a value of nineteen point six 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 twenty-eight point three 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 which hold in the question that that value is twenty-eight point three. 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 twenty-eight point three. 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 twenty-eight point three 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 the square rooting both sides of this equation. I now have 𝑟 is equal to the square root of twenty-eight point three 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 twenty-eight point three 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 three point zero zero one three six 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 six point zero zero two seven, 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 forty-nine, so the area is equal to forty-nine 𝜋. 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 forty-nine 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 forty-nine 𝜋 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 the 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 when 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 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 fifteen point seven 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 twelve centimetres with a semi-circle 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 twelve 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 semi-circle, and a quarter circle. Now the square is easy enough, so let’s do that first. Find the area of the square, we just need to do twelve multiplied by twelve. So we have a hundred and forty-four centimetres squared for that part there.

Now let’s think about this semi-circle. 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 twelve centimetres. The radius of the circle, therefore this part here, must be equal to six centimetres. So to work out the area of this semi-circle, 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 semi-circle 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 eighteen 𝜋 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 twelve centimetres. So to work out the area, we’ll do 𝜋 multiplied by twelve 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 thirty-six 𝜋 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 a hundred and forty-four plus eighteen 𝜋 plus thirty-six 𝜋, which gives me an answer of a hundred and forty-four plus fifty-four 𝜋 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 three hundred and thirteen point six 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.