Question Video: Finding the Area of a Sector | Nagwa Question Video: Finding the Area of a Sector | Nagwa

Question Video: Finding the Area of a Sector Mathematics

Work out the area of the given shape, giving your answer in terms of π.

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Video Transcript

Work out the area of the given shape, giving your answer in terms of π.

Whilst it may not look like it, this shape is a sector of a circle. And we know the formula for area of a sector is a half multiplied by π squared multiplied by π, where π is the radius and π is the central angle of the sector in radians. We can use this information to find the area of our sector, but there are two things weβre going to need to do.

First, weβll begin by working out the central angle of our sector. To do this, we use the fact that angles around a point sum to 360 degrees. We know that this little symbol means this angle is a right angle; itβs 90 degrees. To find the size of this central angle then, weβre going to subtract 90 from 360. And that tells us the central angle 270 degrees. However, this formula only works if the central angle is given in radians. So weβre going to use the fact that we know that two π radians is equal to 360 degrees.

We can find the number of radians that is equal to one degree by dividing everything through by 360. And when we do, two π over 360 simplifies to π over 180. So one degree is equal to π over 180. We can now find the size of the central angle of our sector by multiplying the angle in degrees by π over 180. Thatβs 270 multiplied by π over 180, which is equal to three π over two. Now we probably could have spotted this. Remember a full turn is equal to two π. That means half a turn must be equal to one π. We can see that for our central angle, we could complete half a turn and then a quarter of a turn. Thatβs π plus another half π, which we know is the same as three π over two.

We now have everything we need to be able to find the area of our sector. Its radius is five and its angle in radians is three π over two. So the area is a half multiplied by five squared multiplied by three π over two. Five squared is five multiplied by five, which is 25. And when weβre multiplying fractions and integers, itβs sensible to give that integer a denominator. And we give it the denominator of one since 25 is the same as 25 ones. We then multiply the numerators. One multiplied by 25 multiplied by three π is 75π. And when we multiply the denominators, two multiplied by one multiplied by two is four. So the area is 75π over four or 75 over four π. There are no units in the question, so we donβt actually need to give units in the answer. But if we want to, we could say it is equivalent to 75 over four π units squared.

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