Video: Circumferences of Circles

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

14:38

Video Transcript

In this video we’re going to look at how to calculate the circumference of a circle.

Let’s first make sure we know exactly what’s meant by this term circumference. So, the circumference of a circle is the distance all the way around the edge of the circle. So, it’s this distance that I’ve marked in green on the diagram here. It’s what you would refer to for another general 2D shape as the perimeter. But in the case of circles, we have a specific name, circumference, that we give to it.

Before we look at how to calculate the circumference, there are couple of other pieces of terminology that we need to be familiar with. And the first of those is the name given to a line such as the one I’ve drawn here. So, this line crosses from one side of the circumference to the other and it passes through the centre of the circle. And any line that does this is called the diameter of the circle. This is often represented by the letter 𝑑 when we’re doing calculations with circles. So, that’s the first word that we need to be familiar with.

The second word is used to describe the line that starts on the circumference of the circle and joins it to the centre of the circle. So, a line like the one I’ve drawn in orange here. And this type of line is called the radius of the circle. And so, we use the letter 𝑟 when we’re using the radius in calculations to do with circles.

You probably realize that there’s a relationship between the diameter and the radius of a circle. If the diameter starts on the circumference and goes all the way to the opposite side whereas the radius only goes to the centre, then the diameter is twice as long as the radius. So, we have this relationship that the diameter is equal to twice the radius, or the radius is equal to the diameter divided by two if you prefer to think of it that way.

Okay, now, we’re ready to look at how to calculate the circumference of a circle. And there is a formula that we can use to do this. And it’s this formula here. 𝑐, or circumference, is equal to 𝜋 multiplied by 𝑑, where 𝑑, remember, represents the diameter of the circle. Now, if you haven’t met this symbol before, it’s the Greek letter 𝜋, and it’s used to represent a very special number in mathematics. It’s a special number because of this relationship that exists between the circumference and diameter of a circle.

If you were to draw a circle of any size whatsoever, and if you were to accurately measure the circumference, perhaps using a piece of string, and the diameter of the circle, you would find that they are always related in the same way. So, this symbol 𝜋 represents a number. It’s a very special number. It’s what we call an irrational number.

Which means if you were to write it out in its decimal representation, then it would have an infinitely long string of digits after the decimal point. And there would never be any repeating pattern in them. So, it would just go on and on with no repeating pattern within its digits. Now, your calculator will have a 𝜋 button on it so that you can use that within calculations. But sometimes it’s good enough to know that 𝜋 is approximately equal to 3.14. And you can just use it at that level of accuracy within calculations.

So, there’s our formula. The circumference of a circle is equal to 𝜋 multiplied by the diameter. You may also prefer to have the formula in terms of the radius. So, if you remember that the diameter is just twice the radius, we can replace 𝑑 in this formula with two 𝑟. And this will give us a second formula for the circumference. Circumference is equal to two multiplied by 𝜋 multiplied by 𝑟. So, you can use either of these two versions of the same formula. So, let’s look at some examples.

We have a circle here. And would like to calculate the circumference of this circle.

So, looking at the diagram, we can see that the diameter of the circle has been drawn on and has been given to us as 10 centimetres long. So, we need to recall our formula for the circumference of a circle. And I’m going to use this version, that the circumference is equal to 𝜋 multiplied by the diameter. So, all we need to do is substitute the value of 10, which is the length of the diameter, into this formula. So, I have that the circumference is equal to 𝜋 times 10.

Now, you’ll often see that written rather than 𝜋 times 10, you’ll see it written as 10 𝜋. And sometimes you’ll be asked to leave your answers in this form here. Now, that’s an exact value where you haven’t had to do any rounding. And it also means you can perform calculations with circles when you haven’t actually got a calculator if you leave your answer in terms of 𝜋 like this here. But for this question, we’ll go further. We’ll actually evaluate this.

So, on my calculator, I’ll type 10 multiplied by 𝜋. And it will give me this answer of 31.415926, and so on. So, I’ll round it to one decimal place. And that gives me an answer of 31.4 centimetres, to one decimal place. Just notice the units that we use here. Circumference is just a length, so the units, centimetres, are the same as the units that we had for the diameter. Okay, let’s look at a second example.

So, we’d like to calculate the circumference of this circle here.

And looking carefully at the diagram, we can see we haven’t been given the diameter this time. We’ve been given the radius, as that line only reaches the centre of the circle. So, I’ll use the version of the formula that involves the radius. And here it is. The circumference is equal to two times 𝜋 times 𝑟. So, I just need to substitute 7.2 as the radius within this formula.

So, we have circumference is equal to two multiplied by 𝜋 multiplied by 7.2. Now, there are different ways that I can express this. I could express as 14.4𝜋. Or I could express it using a fraction as 72𝜋 over five. Either of those would be absolutely fine. But I’ll go on and evaluate it as a decimal. And so, that gives me an answer of 45.2 millimetres, again rounded to one decimal place. Units, remember, because it’s a length, are millimetres, the same as the units that we used to give the radius.

So, when you’re calculating the circumference of a circle, you just need to be clear first of all. Have I been given the diameter? Have I been given the radius? And depending on which you have, it will then affect which version of the formula you’re going to use. Now, let’s look at a different type of question.

So, the question says, the circumference of a circle, correct to one decimal place, is 32.7 centimetres. Find the radius of the circle, also correct to one decimal place.

So, this is an example of the type of question where we’re actually working backwards from having already been given the circumference, working back to the measurement for the radius. So, we’re going to need the formula for the circumference of a circle. And seeing as the question asks about the radius, I’m gonna start with that version, which was that the circumference was equal to two 𝜋𝑟.

Now, it tells me in the question that the circumference is also equal to 32.7, so I can write down a relationship between these two things. So, I know that two 𝜋𝑟 must be equal to 32.7. Now, it’s a case of working backwards in order to work out what the radius is. So, 𝑟 is on this side of the equation, but it’s currently multiplied by two 𝜋. If I want to get just one 𝑟, then I need to divide both sides of this equation by two 𝜋. So, that will give me 𝑟 is equal to 32.7 over two 𝜋. Then, I can go on and evaluate this using my calculator.

And it gives me 𝑟 is equal to 5.2043 and so on. The question asked for this value to one decimal place, so I need to round it to one decimal place. So, I have 𝑟 is equal to 5.2 centimetres, to one decimal place. Because 𝑟 represents the radius, it represents the length, I’ve got units of centimetres, which were the same as the units of the circumference of the circle. So, this is quite a common type of question where you might be given the circumference and asked to work backwards, either to calculate the radius or to calculate the diameter of the circle.

Okay, let’s look at another question.

This question says using 3.14 as an approximation for 𝜋, calculate the total perimeter of the shape shown.

So, the first thing to note is we’re not using the full decimal value of 𝜋, we’re just using 3.14 as an approximation. So, in our calculations, wherever we would have used 𝜋 before, we’ll use this value 3.14 instead. Now, the shape that we’re looking at isn’t a circle. It’s composed of semicircles. So, we’re not asked for the circumference, we’re asked for the perimeter. So, we need to make sure we look carefully at what the perimeter is composed of.

So, if we start at one point and trace around the shape, we have a semicircle first of all and then a second semicircle. We then have a straight portion here, a third semicircle, and then another straight portion here. So, we need to make sure we include all of those parts in our calculation of the perimeter.

So, let’s think about the semicircles first of all. We’re given this measurement of 18 centimetres, which is the total distance across this shape. So, that distance, if we look at this part here, that distance is twice the diameter of each of the semicircles, which means the diameter of the semicircles must be nine centimetres. Let’s look at calculating the length of the curved parts first. Now, these curved parts aren’t the full circumference of the circle. They aren’t referred to as a circumference. They’re referred to as an arc instead, so we’ll use the term arc length to refer to these.

So, the circumference of a circle would be 𝜋 multiplied by the diameter, but each of these are only half of a circle. So, we’ll do 𝜋 multiplied by nine, but then we’ll halve it, as we only have half of that circumference. So, we have 𝜋 times nine over two, which means that each of these arcs is equal to 4.5𝜋. So, each of these lengths is equal to 4.5𝜋. However, remember the question said we should be using 3.14 as an approximation for 𝜋. So, actually, instead of 𝜋, I should just be using that.

So, I have 4.5 multiplied by 3.14, which gives a value of 14.13 centimetres for each of these arc lengths. Now, remember, there are three of them, so in my final calculation, I’ll have to use that value three times. Now, I mustn’t forget about these two straight portions here. Now, each of those are half the diameter of the circle, so they’re each four and a half centimetres. But as there are two of them, then the total contribution from those part is nine centimetres.

So, now, I need to put all of this together in order to calculate the perimeter. So, the total perimeter is three lots of 14.13, for those three separate semicircular arcs, and then four and a half and four and a half for each of the two straight parts. So, this gives me a total perimeter of 51.39 centimetres for the whole shape. So, two things in this question. If you’re faced with more complicated shape than just a circle, make sure you trace all the way around the edge so you know all the different parts that make up the perimeter. And secondly, if you’re asked to use 3.14 as an approximation for 𝜋, then whenever you have 𝜋 in your calculation you could just replace it with 3.14 instead.

Okay, the final question then in this video.

A bicycle wheel has a radius of 35 centimetres. How far does Alice cycle on her bicycle if the wheel revolves 250 times?

So, I always think it’s helpful just to sketch a very quick diagram to understand the situation first. So, Alice’s bicycle here is represented by a circle. And I were told has a radius of 35 centimetres. So, to answer this question, first of all we need to work out the circumference of the bicycle wheel and then we need to multiply it by 250, as on this journey it revolves 250 times.

So, circumference, remember, is equal to two 𝜋𝑟. So, we’ll be substituting the value of 35 in as the radius here. So, we have circumference is equal to two multiplied by 𝜋 multiplied by 35, which gives us a value of 70𝜋 for the circumference of the bicycle wheel. Now, I’m going to leave it like that for now because that’s an exact value.

Now, we need to work out the total distance travelled. So, if the wheel revolves 250 times, then we need to multiply this value by 250. So, 250 times 70𝜋, which gives us 17500𝜋. Now, I’ll work out what that is as a decimal value. So, it’s 54977.8 something centimetres. Now, given that this is a distance, and we’re talking about someone travelling on a bicycle, it perhaps makes sense to convert that to more reasonable units than centimetres. So, I’ll convert it to metres by dividing by 100. So, we have an answer of 549.7787 metres. And if I round that, perhaps to the nearest metre, then I have an answer of 550 meters, to the nearest metre.

So, there you have it. We’ve seen how to calculate the circumference of a circle from the radius or diameter. We’ve seen how to work backwards from having been given the circumference to calculate the radius or diameter. We’ve seen using 3.14 as an approximation for 𝜋. And we’ve seen giving answers as multiples of 𝜋, or as decimal values.

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