Question

What is the area of the sector of a circle with central angle `(5pi)/2`...?

Answer 1

If we assume that the angle stated is `(5pi)/2` (and not `5pi/2`), then we can see that it is equal to `pi/2`, which is `1/4` of a circle, and the area of the sector, then, is: `(pir^2)/4`

Explanation:

I am assuming that you meant the central angle to be `(5pi)/2` and not `5pi/2`.
If this is the wrong assumption, please let me know.

The first issue with this angle of `(5pi)/2` is that it goes beyond a full circle. A full circle has an angle of `2pi`.
In fact, that angle `(5pi)/2` is equivalent to a full circle `((4pi)/2)` and then another `pi/2` .

So, I interpret the area of interest here to be delimited by that angle of `pi/2`, which is `1/4` of a full circle. The area of a whole circle is: `A_c=pir^2`
Hence, the area of the sector, is `A_s=(pir^2)*(1/4)=(pir^2)/4`

Answer 2

The philosophical issue is whether a sector bigger than `2pi` has an area bigger than the area of the circle. I'm going to say it does, so the area is

`A= (theta/{2pi}) pi r^2 ={theta r^2}/2 = {5pi r^2}/4`