How to prove that focal length is half of radius of curvature in spherical mirrors.

  Let C = center

  F = focus

  θ = angle of incidence of light ray.

A ray coming parallel to the axis goes through focus of the mirror. Then, CM will be perpendicular to mirror. Draw MD perpendicular to CP.

Now, from figure,

Tan θ =MD/CD and tan2 θ = MD/FD

For small θ, tan θ ≈ θ and tan 2 θ ≈ 2 θ.

Then, MD/FD = 2 MD/CD

FD = CD/2

Since, FD = f(focal length), CD =R (radius),

f = R/2.