 Let us learn on how we can arrive at the formula for the area of the sector. Let me just write it here Area of the sector What is a sector the sector is something that you see here It's like the piece of the pizza which looks like this and has some angle theta at the center And we are interested in finding out the area of it. So this pink shaded region So how do we go about this? We will divide the given sector into more sectors And we are going to look at the zoomed version of one small sector like that. This is how This orange sector will look like now Why have we divided the bigger sector into smaller sector is because more divisions that we make this sector Resembles a triangle more and more now this length here is the radius of the circle This is our and this is also our this is just a zoomed version, right? That's why it's bigger. Now if this resembles a triangle Then we can find the area of the triangle if we know the height of the triangle and the length of the base of the triangle Just note that if we make more and more divisions the size of the pie will reduce and the height of the triangle Will actually resemble the radius of the circle, which is that what about the base the base here is Equal to say the one and now this L1 is As good as the curved length of this small pie Right. So how can we get this area one for the orange sector? And it will be half times base, which is L1 times The radius now if I mark L2 L3 and L4 for rest of the sectors I'll get a2 is equal to half times L2 times R a3 as half times L3 times R a4 as half times L4 times R now I have drawn just four divisions But there could be many divisions and if I'm interested in approximating the area of the sector I can write it as a1 plus a2 plus a3 plus a4 and so on if there are more sectors If I do some math over this I can write the total area of the sector as a and write all these terms like this And half and R are the variables which are appearing in every term We can take them common and write them outside and inside the bracket We can simply write L1 plus L2 plus L3 plus L4 and this is nothing but the total length of the arc here From say point a2 point b through point x. Let me just write it x here This is the total length and let's say this total length of the arc AXB is L So the total area would be given by half times R times L Now this total curved length L is a part of the circumference and the length of this Small arc depends upon what angle it subtends at the center We stated at this start that this sector made a theta degrees of angles at the center Total circumference subtends 360 degrees at the center, right? So 2 pi r subtends 360 degrees and if we are interested in finding out what this L is then we can find the ratio of theta and 360 degrees And if we want to find out L from here, we'll just cross multiply So L would be equal to theta divided by 360 degrees times 2 pi r This is basically what we have done 2 pi r times theta divided by 360 degrees. We can replace this value of L here So let's do that If we if we write that down here, we get theta by 360 degrees times 2 pi r If we simplify this we can cancel these 2 and 2 and we are left with theta divided by 360 degrees times pi r times r Which is pi r squared and this is the area of the sector Now let us solve a quick problem. Let us say the radius of the circle is 14 centimeter and the theta here is 120 degrees. Now, what is the area of the sector oaxb from the formula of the area of the sector? We know that area is theta by 360 degrees times pi times r squared So here area would be theta, which is 120 degrees divided by 360 degrees times pi We will use the value 22 by 7 for pi and times r squared, which is 14 times 14 Now if we solve this we get area to be equal to 1 by 3 which is equal to 120 by 360 times 22 and if we just divide 14 by 7 we get 2 So 22 times 2 times 14, which gives us the area as 616 by 3 square centimeters or 205.33 square centimeter