EF is perpendicular to the common line of non-planar segment AB and CD, so the center of the ball is on EF.

Let the center of O be e, x and F(4-x).

Connect OA and OC, then OA=OC=R= spherical radius.

Pythagorean theorem:

AE？ +EO？ =OA？ =OC？ =OF？ +CF？

4? +x？ =(4-x)？ +2?

16+x？ = 16-8x+x？ +4

8x=4，x= 1/2

r？ =4? + 1/2? = 16+ 1/4=65/4

Sphere area =4πR? =4π×65/4=65π