What is the relationship between the radius and the focal length of the mirror?

The focal length of a spherical mirror is equal to half of its radius of curvature
{f = 1/2 R}

Proof: In below figure 1 and 2, a ray of light BP' travelling parallel to the principal axis PC is incident on a spherical mirror PP'. After reflection, it goes along P'R, through the focus F, P is the pole and F is the focus of the mirror. The distance PF is equal to the focal length f. C is the centre of curvature. The distance PC is equal to the radius of curvature R of the mirror. P'C is the normal to the mirror at the point of incidence P'

For a concave mirror:
In figure,

∠BP'C	=	∠P'CF (alternate angles)
and ∠BP'C	=	∠P'F (law of reflection, ∠i = ∠r)
Hence ∠P'CF	=	∠CP'F
∴ ΔFP'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
	=	1/2 PC
or f	=	1/2 R

For a convex mirror:
In figure,

∠BP'N	=	FC∠P' (corresponding angles)
∠BP'N	=	∠NP'R (law of reflection, ∠i = ∠r)
and ∠NP'R	=	∠CP'F (vertically opposite angles)
Hence ∠FCP'	=	∠CP'F
∴ ΔFP'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
	=	1/2 PC
or f	=	1/2 R

Thus, for a spherical mirror {both concave and convex), the focal length is half of its radius of curvature.

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In optics, we have come across different types of mirrors and have studied their properties in detail like the radius of curvature, focal length, focal point, dimension of the mirror, imaging capacity (erect or inverted), thickness, refractive index and many more. Mirrors are divided into two types:

Plane Mirrors
Spherical Mirrors

In this article, let us understand the relationship between focal length and radius of curvature.

The reflecting surface of a spherical mirror may be curved inwards or outwards. The focal length of a mirror is represented as f and is defined as the distance between the focus and the pole of the mirror. The radius of curvature is represented as R and is defined as the radius of the mirror that forms a complete sphere.

A ray of light AB, which is incident on a spherical mirror at point B and is parallel to the principal axis. CB is normal to the surface at point B. CP = CB = R is the radius of curvature. After reflection from mirror the light will pass through the focus of the concave mirror F or will appear to diverge from the focus of the convex mirror F and obeys the law of reflection i.e. i = r.

From the geometry of the figure,

∠BCP = θ = i (As ∠BCP and ∠ABC are alternate angles)

In ΔCBF, ∠CBF = θ = r

∴BF = FC (because i = r)

If the aperture of the mirror is small, B lies close to P, and therefore BF = PF

Or FC = BF = PF

Or PC = PF + FC = PF + PF

Or R = 2 PF = 2f

Or f = R/2

This kind of relation can be applied for convex mirrors too. In this relation, the aperture of the mirror is assumed to be small.

Radius of curvature is observed to be equal to twice the focal length for spherical mirrors with small apertures. Hence R = 2f .

We can say clearly that the principal focus of a spherical mirror lies at the centre between the centre of curvature and the pole.