Focal Length of a Spherical Mirror

Rays arriving parallel to the axis of a spherical mirror cross the axis at half the radius of curvature. The proof is three lines. Take a ray parallel to the axis striking the mirror at some height; the normal at that point is the radius, pointing back to the centre of curvature $C$. The law of reflection sends the ray away at the same angle to that normal, and the triangle formed by the strike point, $C$, and the place where the reflected ray crosses the axis has two equal angles — so it is isosceles, and the crossing point sits exactly halfway between the mirror and $C$. That "exactly" holds only while the angles are small, and everything interesting about mirror design lives in the failure of that assumption.

Grind a telescope blank to a 4 m radius of curvature and you have a 2 m focal length. Run it the other way for a real specification: an f/5 telescope with a 200 mm aperture needs $f = 1000$ mm, so $R = 2000$ mm — and $R$ is what the mirror maker actually measures during figuring, by finding the point where the mirror images a pinhole back onto itself. A domestic case: a concave shaving mirror with $R = 1$ m has $f = 0.5$ m, so a face held closer than 0.5 m gets an upright, enlarged, virtual image, and a face held farther away suddenly appears upside down. Most people have noticed that flip without knowing they were crossing a focal point.

The small-angle proviso is the origin of spherical aberration. Rays striking farther out on a sphere cross the axis slightly closer to the mirror, so the focus is a smear rather than a point, and the smear grows as the fourth power of the aperture ratio. Newton's first reflecting telescope of 1668 used a spherical mirror and was small enough that this hardly mattered. Modern primaries are figured into paraboloids, which bring all parallel axial rays to one exact point — at the price of coma for anything off-axis, which is the trade the Schmidt and Ritchey–Chrétien designs exist to manage. Worth keeping in proportion: the deviation between the sphere and the required paraboloid on a large mirror is only a few wavelengths of light, and removing those few wavelengths is the hardest and slowest part of making one.

The sign convention is what people get wrong. A concave mirror has its centre of curvature in front of it, so $R$ and $f$ are positive and it can form real images. A convex mirror — the passenger-side wing mirror, the security dome in a shop — has its centre of curvature behind the reflecting surface, so $R$ and $f$ are negative, and it can only ever form upright, reduced, virtual images. That negative focal length is the whole content of "objects in mirror are closer than they appear": the reduced image reads to the eye as a more distant car. Enter a convex mirror's radius as a positive number and every answer downstream is wrong. Two further cautions: $f = R/2$ is a mirror relation only — a lens with the same surface radii has a focal length given by the lensmaker's equation and depends on the refractive index, and substituting $R/2$ there is a common and badly wrong shortcut. And $R$ is the radius of the sphere the surface is a cap of, not the radius of the mirror's rim.