Consider a concave mirror as shown in the figure above:

A ray of light AB travelling parallel to the principal axis PC is incident on a concave mirror at B.

After reflection, it goes through the focus F. P is the pole of the mirror. C is the centre of curvature.

The distance PF= focal length f.
The distance PC= radius of curvature R of the mirror. 
BC is the normal to the mirror at the point of incidence B.

∠ABC=∠CBF (Law of reflection, ∠i=∠r)
∠ABC=∠BCF (alternate angles) 
which implies ∠BCF=∠CBF
Hence, ΔFBC is an isosceles triangle.
So, sides BF=FC

For a small aperture of the mirror, the point B is very close to the point P,
which means BF=PF
Hence, PF=FC=1/2 PC
In other words, f=1/2 R or R = 2f