 A circle with radius 15 cm has a segment with a central angle of 60 degrees. What is the area of the segment? We have a circle over here and the radius of the circle is given to us as 15 cm. And also this circle has a segment over here we have this yellow shaded region which is bounded by a chord AB and this arc of our circle. So this is a segment and this segment makes an angle of 60 degrees at the centre. So the central angle made by this segment is 60 degrees and we need to figure out the area of this segment, this shaded region. Now how do we go about it? Here in this circle we have the sector OAB right over here. This region is a sector as it is bounded by 2 radii and this arc of our circle. And if we look closely we can see that this sector is actually made up of this triangle OAB which is right over here and this segment over here. So in order to figure out the area of our segment what we can do is we can subtract this triangle from our sector and we will be left with this segment. So let me write this out clearly for you. If I have my sector over here and if I subtract my triangle from this sector I will put a subtraction sign. So if I subtract these 2 areas I will be left with the area of my segment right over here. So why don't you go ahead and figure out the individual areas of the sector and the triangle and do this on your own. We can see over here that the area of this sector is actually a part of the total area covered by a circle. So over here the area of our sector OAB will be pi r squared which is the total area covered by a circle times the angle it makes at the center which is 60 degrees. So pi r squared times 60 degrees divided by the total angle which a circle makes at the center which is 360 degrees. So this is the area of our sector and if you want to learn more about it you can go back and watch the previous videos in which we discussed the area of a sector in detail. Moving on to our triangle OAB right over here the angle made by O at the center is also 60 degrees and also OA and OB being the radii of our circle have equal lengths of 15 centimeters each and as both these lengths are equal our triangle OAB is an isosceles triangle. This means that the angle opposite to the equal sides must be equal so angle A should be equal to angle B and also as angle O is 60 degrees the sum of angle A and angle B would be equal to 180 degrees minus 60 degrees. So this is equal to 120 degrees and as angle A and angle B are equal each of these angles should measure 60 degrees each and as all the angles are 60 degrees our triangle OAB is equilateral triangle and we know that the area of an equilateral triangle is under root 3 by 4 times A square where A is the measure of a side of the triangle. So let's simplify this now over here 60 by 360 is 1 by 6 so this becomes pi times R squared which is 15 times 15 as the radius is 15 times 1 by 6 so 3 times 2 is 6 3 times 5 is 15 and over here we have under root 3 by 4 times 15 squared I will write the other 15 as 5 times 3 and you will know in a second why I did that. Now all we have to do is subtract these two so we can take 15 times 5 common out of these terms so 15 times 5 is 75 and we are left here with pi by 2 minus 3 root 3 we have 3 root 3 by 4. So this is the area of our segment right over here and this is equal to 75 times pi by 2 minus 3 root 3 by 4 centimeters squared.