The lens focal length (f) is the distance between the centre of the lens and the point at which the reflected light meets the centre line of a beam of light moving parallel to the centre line (principal axis).
The curvature radius (r) is the radius of a full sphere that forms the lens.
The red line reflects the light that enters the lens (AB) and that reflects off the lens (BF).
We know that the green line, RB, which reflects the circle's radius line, bisects the ABF angle since the lens is always at the right angle.
The angles of ABR and RBF are therefore equivalent. Because of the law of alternate angles, we also know that ∠ABR is equal to ∠BRF. Therefore the triangle BRF is a triangle of isosceles. The BF and RF lines are therefore equivalent.
We also know that for lenses that are thin, BF and FC are about equal.
Therefore:
RF=BF=FC
RC=RF+FC=FC+FC=2FC
R=2f
Therefore, the radius(R) of curvature is twice the focal length(f).