Can the angle be obtained using just this information? Also, wouldn’t that be nice?
Of course it can 
Assume you cut this triangle in half, exactly on the line from the corner you marked with ? and the center of C. Then you have a right-angled triangle.
C/2 = A * sin(phi/2)
C / (2A) = sin(phi/2)
phi = 2 * arcsin(C / (2A))
That did it, thanks!
Do a google search on the following … you will find lots of formulas for computing similar stuff. Your application in particular is SSS
Side-Side-Side
Side-Angle-Side
Angle-Side-Side
Law of Cosines