Work out the area of the given shape, giving your answer accurate to two decimal places.
The given shape is a sector of a circle. And we recall the formula for the area of a sector of a circle with radius 𝑟 and angle 𝜃 radians to be one-half 𝑟 squared 𝜃. We can see the radius of our circle to be five units. But what about the angle? Well, there are a number of ways we can calculate this angle. We could recall the angles around a point sum to 360 degrees. And since we have a right angle marked on the diagram, we subtract 90 from 360. And we can see that the central angle in our sector is 270 degrees.
Remember that we said our angle needed to be in radians. So we recall the fact that two 𝜋 radians is equivalent to 360 degrees. 270 is three-quarters of 360. So we find three quarters of two 𝜋. That’s three 𝜋 over two radians. So 𝜃 is three 𝜋 over two. And therefore, the area of the given shape is a half multiplied by five squared multiplied by three 𝜋 over two. That’s equal to 75𝜋 over four square units.
We were told to give our answer accurate to two decimal places though. Popping this value into our calculator, we get 58.904 and so on. The second digit after the decimal point is a zero. And the deciding digit is this number four. Remember, if the deciding digit is less than five, we round our number down.
And doing so, we can see that the area of our sector is 58.90 square units.