[Music] in this video we're going to see how to calculate the area of a sector and a length of an arc to quickly recap what these terms mean remember that when we take a slice of a circle like this we get a minus sector contained within a minor arc and the two radii and a major sector contained within a major arc and the two radii and when you get questions about this stuff in the exam they are normally going to give you this angle here of the minor sector and the length of the radius now if you want you can just memorize these two formulas here and by plugging in the angle as x and the radius as aha this will give you the area of a sector or the length of the arc what i want to do in this video though is go through the concept behind the formulas so that they make sense because when we break them down they're much easier to understand and they're also a lot easier to use in tricky questions so for me personally the best way to think about this topic is to think about how big the sector or arc is as a fraction of the entire circle for example if our angle here was 90 degrees then that would mean that our sector was one quarter of the full circle because remember a whole circle is 360 degrees and 90 is one quarter of 360. and so this means that the area of this sector would be one quarter of the area of a full circle and the length of the arc would be one quarter times the circumference of the full circle so to find out the area of our sector we'd just do one quarter times pi r squared because pi r squared is the formula for the area of a circle and then for the length of the arc which do one quarter times two pi r because two pi r is the formula for the circumference so if the radius of our circle in this case were 6 centimeters for example then the area would be one quarter times pi times 6 squared which is 28.3 square centimeters and the length of the arc would be one quarter times two times pi times six which is nine point four two centimeters now this technique works just fine if you've got something easy like a quarter a third or one half of a circle the issue though is that if you have a more complicated angle like say 113 degrees then it's a bit harder to know what fraction of a circle you have in these cases you need to write your fractions with a denominator of 360 because there are 360 degrees in a circle in total and then you just place whatever angle your sector is as the numerator so in this case because the angle of the sector is 113 degrees our fraction would be 113 over 360. or in other words our sector is 113 360 of the circle so if the radius this time was 15 millimeters then to find its area would see 113 over 360 times pi times 15 squared which would be 222 square millimeters and to find the length of the arc we'll do 113 over 360 times 2 times pi times 15 which is 29.6 millimeters before we finish let's have a go at this question here using the two equations from the start of the video and remember that the x just means this angle inside the sector also don't worry that here we're only given the sector not the entire circle sometimes they'll just give you this sort of shape and they will tell you that it's part of a circle or they'll give you the radius and if they give you the radius you know it must have come from a circle so for part a it's asking us to find the area of the sector o a b and o a b is just a way of describing the sector on the left because o a and b are points around the shape so we're going to use our equation area of sector equals x over 360 times pi r squared which if we sub in the 70 degrees and the 14 will be 70 over 360 times pi times 14 squared which will give us 120 square centimeters then for the length of the arc a b so just the arc in between a and b we do x over 360 times 2 pi r which will be 70 over 360 times 2 times pi times 14 so 17.1 centimeters anyway that's everything for this video so i hope that was helpful for you if you want to practice questions on this or anything else in science or maths then head over to our revision site which you can access by pressing the link in the top right corner of the screen otherwise have a fantastic day