Video: Finding the Area of a Sector

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 𝑟 squared multiplied by 𝜃, where 𝑟 is the radius and 𝜃 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 we’re going to need to do.

First, we’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; it’s 90 degrees. To find the size of this central angle then, we’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 we’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. That’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. That’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 we’re multiplying fractions and integers, it’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 don’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.