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derive the formula f=r/2

Consider a ray AB parallel to the principal axis and incident at the point B of a concave mirror. After reflection from the mirror, this ray passes through its focus F, obeying the laws of reflection. If C is the centre of curvature, then CP = R, is the radius of curvature and CB is normal to the mirror at point B.
                   
According to the law of reflection,
                        ?i = ?r
As AB is parallel to CP, so

                        ? =  ?i    (Alternate angles)

                       ? = ?r

Thus  BCF is isosceles. Hence CF = FB.

If the aperture (or size) of the mirror is small, then B lies close to P, so that,FB = FP
Therefore FP = CF = 1/2 CP
or f = R/2
Focal length = 1/2 x Radius of curvature