To make the circle has an area of 4, as much as the square 2x2 in this picture below, what length of radius is needed?

It doesn't take a genius to figure that out.

The area of a circle is given by pi*r^2.

You want that equal to 4, so you want r=sqrt(4/pi)=1.1283791671...

AND at what angle does the intersection (circle line and square line) occur? (North=0°)

Again, doesn't take a genius.

Cos(a)=x/r (or as you have decided to take north to be 0=y/r).

Sticking in the numbers, that means:

a=acos(1/sqrt(4/pi))=27.5971126°

Any more of your very simple homework that you need doing?