Define focal length of a concave mirror. Prove that the radius of curvature of a concave mirror is double its focal length.

The distance between the focus and pole of the mirror is called the focal length of the mirror.

Prove R = 2f 

Let F be the focus, P is the pole of the mirror R be the radius of curvature of the mirror and 'C' be its centre of curvature. 

Let a ray parallel to the principal axis striking the mirror at M. Then CM will be perpendicular mirror at M. Let $\mathrm{\theta }$ be the angle of incidence and MD be the perpendicular from M on the principal axis.

Then from triangle MCP, $\tan \mathrm{\theta }=\frac{\mathrm{MD}}{\mathrm{CD}}\approx \frac{\mathrm{MD}}{\mathrm{CP}}$
$\text{When }\mathrm{\theta }\text{ is small }\mathrm{\theta }=\frac{\mathrm{MD}}{\mathrm{R}}$
From triangle MFD, $\tan 2\mathrm{\theta }=\frac{\mathrm{MD}}{\mathrm{FD}}\approx \frac{\mathrm{MD}}{\mathrm{FP}}$
$\text{When }2\mathrm{\theta }\text{ is small, }2\mathrm{\theta }=\frac{\mathrm{MD}}{\mathrm{f}}$
$\therefore 2\frac{\mathrm{MD}}{\mathrm{R}}=\frac{\mathrm{MD}}{\mathrm{f}}\Rightarrow \frac{2}{\mathrm{R}}=\frac{1}{\mathrm{f}} \therefore \mathrm{R}=2\mathrm{f}$