Here in this video, we will learn about circles, arcs and sectors, including how to find the area and circumference of a circle and how to find the area and arc length of a sector. Circles are round plane figures whose boundaries consist of equidistant points from a fixed point in the centre.
Each part of the circle has a specific name and properties.
So in this lesson, we are going to be working with circles, sectors and arcs. And we're going to be identifying the parts of the circle, finding the circumference of a circle, finding the area of vector and finding the length of an arc. So, parts of a circle, the first thing we're going to look at is the circumference, which is basically the perimeter of a circle or the length around the entire circle. The radius is basically the length between the centre of the circle and the circumference. And the radius is the same throughout the circle.
The diameter is basically the length between one end of the circumference to the other. That line has to go through the centre. A chord is basically going from one end of the circumference to the other. But that does not have to go through, or does not actually go through the centre. A tangent is basically a straight line that crosses the circle at one point.
It does not go through the circle, it SKIMS past it and it crosses it here at that one point. So that's called a tangent, a sector. If you imagine a circle being a pizza, a sector is a slice of pizza. It's essentially a portion of the area taken out and that is always the centre of the circle, meaning these parts here, that's the radius. Now, a segment is part of a sector and it's found between the arc and a chord within a sector.
And the arc is just basically part of the circumference. So not the full length around, just part of it. So those are the parts of a circle. So let's begin by working at the circumference of a circle. Circumference is basically the perimeter of a circle and is found by calculating pi times the diameter for length across or double the radius times pi, which makes sense because double the radius will keep the diameter anyway.
So if we go through an example here, the circle has a radius of 13. Find its circumference to one decimal place. So I'm going to use the radius formula. If you do use the diameter formula, that's also okay, it's just you would need to double your radius first. So plug in or substitute in your value for the radius.
Plug into your calculator two times pi times 13 and that will give you 26 pi, which gives you 81.68 one. And the question asked us to round that to one decimal place, so it's 81.7 and the unit was here in metres. Make sure that you have the correct unit. We're only dealing with length here perimeter. So it would not be metres square or metres cube.
It is just metre.
Okay, area of a sector. So to be able to find the area of a sector, we need to understand how to find the area of a circle. So the area of a circle is found by doing pi R squared or pi times radius squared. Now, remember that a sector is part of the area and it depends on what the degrees are for that sector in order to work out how much of that sector we are looking at. So if, for example, we were looking at a quarter of a circle, a quarter of the full circle, which is 360, would be 90.
So you would substitute a 90 into there. If the angle within a sector, for example, is 60 degrees, then you would put 60 x 360. And what this formula basically means is it's the proportion times pile squared. So it's part of the full area.
So find the area of the minor sector. The minor sector is the smallest sector because we've got the small sector with 60 degrees and then we've got the major sector which is the outside one, which would be 300 degrees. They've specified the minus sector here and they've also asked us to leave our answer in terms of pi. So they don't want us to do any kind of grounding. So let's put our formula on the screen.
The angle we're given a 60 degrees and the radius that we're given is three. So I'm going to substitute that into the formula 60 over 360 times pi times three squared. Plug that into your calculator and it should give it to you already in terms of pi, which is three over two pi or one five pi. You won't lose any marks for leaving as a decimal, but what I would suggest is fractional only because sometimes these fractions can be quite awkward as a decimal. So if you do end up doing it, you don't want to round it, it wants it exact.
And also because leaving in terms of pi, having a random really long decimal with pi is just a bit messy. So the fraction makes it nice and compact.
Okay? And then the final part is arc length. So remember that the arc is basically part of the circumference. So we learnt that the circumference of a circle is pi times diameter. So the arc length, again, it depends on the angle within that spectrum.
So it's the angle over 360 times the circumference formula.
So here we've got the exact time to find the exact length of the minor arc. This is the minor arc again, looking at that sector. And so we're going to plug in 60 degrees and we're going to plug in the diameter which was double the radius. If you use two pi R as your formula for the circumference, that's fine. You would then have 60 or 360 times two times pi times three.
So it depends on which formula you use. I always say for given, if you stick to one, it's usually easier, but it doesn't really make a difference. I've left it in terms of pie. They haven't specified what to leave it out. So you can leave it in terms of pie.
You can also leave it till one DP now, if it doesn't state what surrounded you and you end up rounding, you should always state your rounding. So, for example, if you did do it to one DP, then make sure you put in brackets that you did that to one DP.
Book a lesson with this tutor
Sanaa is currently Deputy Head of Maths at a secondary school and has been tutoring GCSE Maths students for over 10 years. She tailors her approach so that no student is left behind.