For a concave mirror:
In figure (a),
∠BP′C=∠P′CF (alternate angles) and
∠BP′C=∠P′F (law of reflection,∠i=∠r)
Hence ∠P′CF=∠CP′F
∴ FP′C is isosceles.
Hence, P′F=FC
If the aperture of the mirror is small, the point P′ is very close to the point P,
then P′F=PF
∴ PF=FC=PC2
or f=R2
For a convex mirror:
In figure (b),
∠BP′N=∠FCP′ (corresponding angles)
∠BP′N=∠NP′R (law of reflection, ∠i=∠r) and
∠NP′R=∠CP′F (vertically opposite angles)
Hence ∠FCP′=∠CP′F
∴ FP′C is isosceles.
Hence, P′F=FC
If the aperture of the mirror is small, the point P′ is very close to the point P.
Then P′F=PF
∴PF=FC=PC2
or f=R2
Thus, for a spherical mirror {both concave and convex), the focal length is half of its radius of curvature.