Notes 11.5 Area of a Circle and Sector of a Circle

[00:00:00]hey guys today we're going to talk about notes 11.5 which is on the area of a circle and the sector of a circle starts off we're going to go ahead and talk about that area of a circle now you might notice there is a small typo in that it's a type equation here that's exactly where we are going to write the area of the circle so that area formula is the area is pi times the radius squared and that's all you gotta do um and of course your radius of a circle is goes from the center of the circle to a point on the circle is that distance there so let's go ahead and try some examples with this example number one says to find the area of a circle with a radius of nine inches all right here we go again i'm gonna write the formula down area is pi times the radius squared this one tells you that radius is nine so we're gonna go ahead and do pi times 9 squared and as you know 9 squared is 81 so we can write this as 81 pi and our units on this because we're talking about an area now are back to be inches squared or square inches you can also ahead go ahead and get a decimal approximation for this we would just go ahead and use 3.14 for our pi value and do 81 times 3.14 and as we run that calculation through a calculator we get down to 25.8 inches squared you'll find example two to be very similar to example one so you might do that one now uh because it'll be found in your quick check let's keep on going example number three says to find the radius of a circle if the area is 225 feet squared 225 pi feet squared this problem is kind of working backwards here because i've given you the area and we're going to be finding the radius so again just reminder our formula is area that's pi times the radius squared and we can go ahead and plug in what we know the area is 225 pi we do not know the radius i'm going to leave this as pi times the radius squared first step here is to isolate your variable so i'm going to go ahead and divide both sides by pi as you do this it's actually very nice because pi cancels on the right it cancels on the left so we're just left with this 225 is equal to the radius squared now again just a brief reminder on how you want to get the radius from here because you have a squared term the only way to get down to it is to go ahead and take the square root of both sides as we take that square root of r squared we do get r our radius as we take the square root of 225 you might need calculate for that but it is 15. so this tells you your radius is 15 feet all right you'll find example four very similar to the one i've just done here use mine as an example let's keep on going i'm going to go ahead and talk about the area of the sector of a circle so right here we say that a sector of a circle is a region of a circle bounded by a central angle two radii and its intercepted arc in the diagram below the sector is the shaded region and if you want to put some more like realness to this all right a sector of a circle looks a lot like the slice of a pizza or a slice of pie all right so it is this red region here so you see some other things i have labeled here s is going to be our area of a sector x degrees is going to be our central angle and r is going to be our radius so we can go ahead and find the area s of a sector it's determined by the central angle to be the central angle divided by 360 times pi times radius squared okay so that's all there is to this formula let's go ahead and try some examples looking at example number five it says to find the area of the shaded sector and be very careful here i went ahead and i gave you the angle that's a 150 degrees but that is not the shaded sector so we're going to need to figure out the angle of the shaded sector and that is pretty easy we know the total for a circle is 360 degrees so the red part here let's go ahead and grab my red pen the red part here is going to be 360 degrees subtract off the 150 that's already shown and of course if we do that subtraction there we're going to get 210 degrees so the angle of this red part here is going to be 210 degrees that's what we're going to need for our formula so our formula again is that the area of a sector is given by the angle divided by 360 times pi times r squared all right well our angle is that 210 so we can go ahead and say 210 over 360 times pi and your radius here is five so we're gonna have times five squared if you run these calculations on your calculator again leaving the pi out of it for just a second here if you just do 210 times 5 squared or 210 times 25 and then divide by 360 you'll get down to 14.6 pi or if you just run the whole thing through your calculators all at once here we'd be doing 14.6 times the 3.14 eventually and that would get us down to this 45.8 and again this is a big or here you can do either one if somebody asks you to leave your answer in terms of pi that's your answer somebody asked for a decimal approximation that's your answer it just depends on what you're looking for then you're going to find go ahead and find out uh the next two examples are both found in your quick check i've left you a couple hints here uh the first sin says be careful the shaded part does not have a central angle of 75 degrees very similar to the one i've just done and example number seven is to find the area of the shaded region and be careful what does that symbol mean you've seen it a lot this year but what does that symbol mean for the central angle moving on to our next example we have example number eight find the length of the radius if the area of the sector is 13.5 feet squared and the central angle is 135 degrees all right so we're going to go ahead and write that formula down again the area sector s is given to be the angle divided by 360 times pi times the radius squared let's plug in what we know we know the area of the sector is 13.5 pi and we know the angle is 135 degrees and we're looking for that r value that's going to be our unknown all right guys we saw this in the previous lesson when you have this 360 here the very first thing you're probably going to want to do is actually to in a sense cross multiply get that 360 to the other side and so it's going to look something like this we're going to have our 13.5 times pi times that 360. and again i'll just reiterate from the previous lessons when we think about cross multiplication we can think about this all over one and really we thought about that whole thing as a numerator over there and so when we do that cross multiplication we just have one times that numerator which is just going to leave us with that numerator 135 times pi times the radius squared all right first thing i'm going to go ahead and do is recognize that i've got 135 pi next to my radius and i don't like that and so i'm going to go ahead and move into dividing by this 135 pi on both sides all right and um i guess i've kind of jumped a couple steps here but we're doing all right still as we do this all right this would cancel and it would leave us with this r squared on the on the right and what i want to do on the left side is i'm actually going to do this in two steps here because i kind of jumped here i'm going to go ahead and multiply the 13.5 times the 360.

[00:07:09]as i do the 13.5 times the 360 i get down to 4860 pi is still there and it's still divided by 135 i'm just going to leave that on there right now for just a second here some of you guys are probably already screaming at me look at those pies there's one on the top there's one on bottom we can cancel those out and then of course now all i need to do is do that 4860 divided by the 135 and grab my calculator type that in as you type that in that division gives you 36 and as we saw in the previous problem we do need to get r by itself and once we have an r squared we can find r just by taking a square root of both sides and a lot of you out there probably do know what the square root of 36 is the square root of 36 is going to be 6.

[00:07:50]so your radius is 6 feet and again just my friendly plug you do not need to put the units into it's learning just the numerical values for your answer and that's all we need so it's six feet uh for your radius here as you look at example number nine you'll find it's very similar to example eight so please use mine as a reference we're gonna move on to our last one together here which is example number 10. it says to find the measure of the central angle of a sector if the area of the sector is 8 pi centimeters squared and the radius is 8 centimeters all right so we're going to keep using that formula that we got from the very beginning our area of the sector is given by the angle divided by 360 times pi times the radius squared this time we're going to be solving for the angle because they've given us the area of the sector so that s that give us the radius which is that eight let's plug everything in area the sector is eight pi angle is unknown i'm going to keep using the word angle for my variable on that divided by 360 times pi times the radius squared let's simplify here and um let's go ahead and actually move that 360 to the other side next let's go ahead and deal with that as we've previously done we have 8 pi times our 360. it's going to equal the angle times pi and 8 squared let's go ahead and just call that 64.

[00:09:15]all right so as we kind of recognize here our variable is this word angle and we don't really want the pi or the 64 next to it so let's go ahead and divide that out as we divide out this 64 pi dividing on both sides here doing something very similar to the last problem here where on the right side i will be left with the angle and on the left side i have a lot to simplify here i'm going to start by going ahead and multiplying the 8 and the 360. eight times 360 will get you 2880 pi still over that 64 pi that we just divided by and if you weren't screaming at me last time hopefully you're screaming at me now those pies are going to cancel all right those pies are going to cancel right off the bat and now we just have to go to our calculator and figure out what 280 is divided by 64. so our angle once we do that on our calculator we get to 45 degrees to find example 11 very similar to example 10 please do minus the reference that wraps it up for this one though so if you guys have any other questions don't be afraid to reach out