Spherical mirrors

A dentist's tiny curved mirror, the wide convex mirror at a blind road turn, and a car's headlight reflector are all spherical mirrors — mirrors whose reflecting surface is a cut-out piece of a hollow sphere. One curves inward and gathers light to a point; the other curves outward and spreads light apart. Before any ray diagram, mirror formula, or magnification rule can make sense, you need the vocabulary that describes a spherical mirror's own geometry: its pole, its centre of curvature, its principal axis, and its focus. This article builds that geometry from scratch, ties it together with the single relation $f = R/2$ , and works it both ways — from radius to focal length and back — so the terms stop being separate flashcards and become one connected picture you can sketch on demand.

The core insight

A spherical mirror's focal length is always half its radius of curvature, $f = R/2$ , because the principal focus is the point where paraxial rays reflected off the mirror converge (or appear to diverge from), located geometrically midway between the pole and the centre of curvature.

The core idea

A spherical mirror is a mirror whose reflecting surface is part of a hollow sphere. There are two kinds, distinguished by which side of that sphere reflects light. A concave mirror curves inward, like the inside of a spoon. Rays of light travelling parallel to its axis are brought together — converged — at a single point after reflection, which is why it is also called a converging mirror. A convex mirror curves outward, like the back of a spoon. Parallel rays are spread apart after reflection, so it is also called a diverging mirror. Every spherical mirror carries the same set of landmarks, all defined with respect to the sphere it is cut from:

Pole ( $P$ ) — the centre point of the mirror's reflecting surface, the natural origin for measuring distances along the mirror.
Centre of curvature ( $C$ ) — the centre of the hollow sphere the mirror is part of. For a concave mirror $C$ lies in front of the reflecting surface (on the same side as the light source); for a convex mirror $C$ lies behind it.
Radius of curvature ( $R$ ) — the radius of that sphere, equal to the distance $PC$ .
Principal axis — the straight line through $P$ and $C$ , extended on both sides; it is the reference line for every ray diagram.
Principal focus ( $F$ ) — the point on the principal axis where rays travelling parallel to the axis converge, for a concave mirror, or appear to diverge from, for a convex mirror, after reflection.
Focal length ( $f$ ) — the distance $PF$ , from the pole to the principal focus.
Aperture — the diameter of the mirror's reflecting surface, i.e. the usable, illuminated part of the sphere.

Figure

A concave mirror with pole P centre of curvature C and focus F marked on the principal axis, and a ray parallel to the axis reflecting through F

For a mirror of small aperture, the focal length and the radius of curvature are always related by:

f = \frac{R}{2}

where: $f$ is the focal length (the distance $PF$ ) and $R$ is the radius of curvature (the distance $PC$ ). The relation holds for both concave and convex mirrors — the sign convention that decides whether $f$ and $R$ count as positive or negative is treated separately. It follows purely from the geometry of reflection for rays that travel close to the principal axis, called paraxial rays, and it places $F$ exactly midway between $P$ and $C$ .

Derivation AWhy f equals half of R

The goal here is to show, using nothing more than the law of reflection and the geometry of a circle, why the focal length of a spherical mirror must equal exactly half its radius of curvature for rays travelling close to the principal axis, so that the relation $f = R/2$ is not simply a fact to memorise but a direct consequence of how a paraxial ray bounces off a curved surface and crosses the axis.

Step 1. Consider a ray travelling parallel to the principal axis that strikes the mirror at a point $M$ close to the pole $P$ . Since $M$ lies on the sphere of radius $R$ centred at $C$ , the line $CM$ is a radius of that sphere, and a radius is always perpendicular to the sphere's surface at that point — so $CM$ is the normal to the mirror at $M$ . Because the incoming ray is parallel to the axis, the angle it makes with the normal $CM$ equals the angle the normal itself makes with the axis; call this angle $\theta$ .

Step 2. By the law of reflection, the reflected ray leaves $M$ making the same angle $\theta$ with the normal $CM$ , on the other side of it. In the triangle formed by $C$ , $M$ , and the point $F$ where the reflected ray crosses the axis, the angle at $C$ is $\theta$ and, because the triangle $CMF$ is isosceles near the axis, the exterior angle at $F$ — the angle between the reflected ray and the axis — works out to $2\theta$ , exactly double the angle at $C$ .

Step 3. For a paraxial ray, $M$ sits very close to $P$ , so the horizontal distance from $M$ down to the axis is small compared with both $PC$ and $PF$ . This lets the two triangles be read as nearly flat: the angle at $C$ obeys $\theta \approx \dfrac{PM}{PC}$ and the angle at $F$ obeys $2\theta \approx \dfrac{PM}{PF}$ , where $PM$ is the (small) height of the strike point above the axis.

Step 4. Divide the second relation by the first. The common factor $PM$ cancels, leaving $\dfrac{2\theta}{\theta} = \dfrac{PC}{PF}$ , so $2 = \dfrac{PC}{PF}$ , which rearranges to $PF = \dfrac{PC}{2}$ . Since $PF$ is the focal length $f$ and $PC$ is the radius of curvature $R$ , this is exactly $f = R/2$ — the principal focus sits at the midpoint of $P$ and $C$ for any paraxial ray, regardless of which point $M$ near the pole was chosen.

Worked example

The radius of curvature of a concave mirror is 32 cm. Find its focal length, and state how far its centre of curvature lies from its focus.

Step 1. Identify the given quantity and the relation to use. The radius of curvature is $R = 32\ \text{cm}$ , and since $f$ and $R$ are connected by $f = R/2$ , this is a direct substitution.

Step 2. Substitute and compute the focal length: $f = \dfrac{R}{2} = \dfrac{32\ \text{cm}}{2} = 16\ \text{cm}$ . Parallel rays striking this mirror near the pole converge at a point 16 cm in front of the mirror, along the principal axis.

Step 3. Locate $C$ relative to $F$ . Since $F$ is the midpoint of $P$ and $C$ , and $PF = 16\ \text{cm}$ while $PC = R = 32\ \text{cm}$ , the distance from $F$ to $C$ is $PC - PF = 32 - 16 = 16\ \text{cm}$ . So $C$ lies a further 16 cm beyond $F$ , along the same axis — as it must, since $F$ is always exactly halfway between $P$ and $C$ .

Answer: the focal length is 16 cm, and the centre of curvature lies 16 cm beyond the focus, 32 cm from the pole.

Competitive stretch

A convex mirror used as a vehicle's rear-view mirror has a focal length of 20 cm. Find its radius of curvature and the distance of its centre of curvature from the pole.

Step 1. Use $f = R/2$ rearranged for $R$ : $R = 2f$ .

Step 2. Substitute $f = 20\ \text{cm}$ : $R = 2 \times 20\ \text{cm} = 40\ \text{cm}$ .

Answer: the radius of curvature is 40 cm, so the centre of curvature $C$ lies 40 cm from the pole — behind the reflecting surface, since this is a convex mirror — twice as far as the focus $F$ , which sits 20 cm behind the mirror.

Common traps

Mixing up concave and convex. Concave mirrors "cave in" and converge parallel light; convex mirrors bulge outward and diverge it. Swapping the two leads to wrong predictions in every ray diagram that follows.
Treating $C$ and $F$ as the same point. They are always different points on the principal axis, with $F$ exactly midway between $P$ and $C$ — never coincident, never swapped.
Trying to derive $f = R/2$ from memory of trig identities under exam pressure. For CBSE Class 10, the relation is to be stated and applied; the geometric justification above is for understanding, not for reproducing step by step in an exam answer.
Forgetting that $C$ is not a mark on the glass. It is the centre of the imaginary complete sphere the mirror is cut from — for a concave mirror it sits in the space in front of the mirror, not anywhere on the reflecting surface itself.

Ask the tutor: If P, C, and F keep blurring together on a diagram, ask the AI tutor to quiz you on labelling a mirror sketch until the positions click.

Why is the focal length exactly half the radius of curvature, and not some other fraction?

It comes from the geometry of reflection for rays close to the axis. When a ray parallel to the axis reflects off the mirror, the law of reflection makes the reflected ray bend through twice the angle that the normal makes with the axis at the strike point. Working through that doubling with the small-angle approximation for a point near the pole shows the crossing point on the axis must sit at exactly half the distance PC — no other fraction is consistent with the doubling. The derivation above walks through this in full.

Does f = R/2 apply to convex mirrors too, or only concave ones?

It applies to both. The geometric argument used to derive it — a normal through the centre of curvature, equal angles of incidence and reflection, and the paraxial approximation — works the same way whether the mirror curves inward or outward. What differs between the two mirrors is not the relation itself but where C and F are located: in front of the mirror for a concave mirror, behind it for a convex mirror, which the sign convention (covered separately) keeps track of.

What exactly counts as the aperture, and why does it matter for this formula?

The aperture is the diameter of the mirror's actual reflecting surface — the physical size of the glass, not any distance along the axis. It matters because f = R/2 is only accurate for paraxial rays, ones that strike close to the pole. A mirror with a large aperture catches rays far from the pole too, and those rays no longer converge at exactly the same point F, causing a blur called spherical aberration. For small-aperture mirrors, as assumed throughout this topic, the approximation is excellent.

Spherical mirrors

The core idea

Worked example

Competitive stretch

Common traps

Before this

Where this leads