Step-by-step Answer:
Yes, there is a way. However, the equation involved is much simpler if we use radians for angles.
A complete circle (360 degrees) equals 2pi radians.
For example, 90 degrees equals 1/4 of 2pi = pi/2 radians, etc.
The area of a sector of radius r subtending an angle of alpha has an area of Area of sector = r^2(alpha)/2
which means that the area of a complete circle (angle alpha=2pi)
Area of circle = r^2(2pi)/2 = pi (r^2) as we all know.
So for given, A=pi/2
Area of sector = r^2(alpha)/2 = 2(alpha) = pi/2, we solve for
alpha = pi/2/2 = pi/4 (which is same as 45 degrees).
Therefore the fraction of the sector
= (pi/4)/ (pi/2) = 1/8
From the above, a simple formula can be found, which is dividing simply the area of the sector by the area of the circle, pi(r^2), i.e. (pi/2)/(pi(2^2))=(pi/2)/(pi/4) = 1/8