Relation Between Radius of Curvature and Focal Length

Let us consider a ray of light AB strikes on a surface of a concave mirror. Here PD is the radius of curvature and PC is the focal length. Assuming that the aperture of a mirror is very small, we can conclude that PC is nearly equal to PB. Here, angle ABD is an incident angle. BD is a normal and angle DBC is a reflected ray.

Now, using simple geometrical laws in the above figure:

∠ ABD = ∠ CBD  (∵ i = r)
∠ ABD = ∠ BDP( ∵ Being alternate angle)
Since ∆ BDC is an isosceles triangle, CB is equal to CD.
Now, PD = BC + CD
or, PD = PC + CD
or, PD = PC + PC
or, PD = 2PC
∴ R = 2F

This is the required relation between radius of curvature and focal length.

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