Last updated: 2026-06-16

The mirror equation

The mirror equation relates three distances measured along the principal axis of a spherical mirror: the focal length $f$, the distance from the object to the mirror $d_o$, and the distance from the image to the mirror $d_i$. They are connected by

Given the focal length and where the object sits, this calculator solves for the image distance and the magnification, then reports whether the resulting image is real or virtual and enlarged or reduced. The same relationship governs everything from shaving and makeup mirrors to the giant primary mirrors of reflecting telescopes.

Sign convention

The formula only works once a consistent sign convention is fixed. Here, distances on the reflecting side of the mirror — the side the light comes from — are positive.

• The focal length is positive for a concave (converging) mirror and negative for a convex (diverging) mirror. It equals half the radius of curvature, $f = \frac{R}{2}$.
• A positive image distance means a real image that forms in front of the mirror, where the reflected rays physically cross. A negative image distance means a virtual image that appears to sit behind the mirror.
• The magnification is $m = -\dfrac{d_i}{d_o}$. A negative value means the image is inverted; a positive value means it is upright. Its size relative to the object is $|m|$.

Formula

Solving the mirror equation for the image distance gives $d_i = \frac{f\,d_o}{d_o - f}$, which is what the calculator evaluates.

Worked example

A candle is placed 50 cm in front of a concave mirror whose focal length is 20 cm. The image distance is

The magnification is $m = -\frac{33.3}{50} = -0.67$. Because $d_i$ is positive the image is real, and because $m$ is negative and smaller than 1 in size the image is inverted and reduced — exactly the upside-down, shrunken reflection you see in a concave mirror when you step well back from it.

Concave versus convex

A concave mirror can form either kind of image. When the object is farther away than the focal point it produces a real, inverted image; when the object is closer than the focal point the image becomes virtual, upright and magnified, which is how a makeup or shaving mirror works. A convex mirror, with its negative focal length, always produces a virtual, upright, reduced image no matter where the object is. That shrinking gives a wide field of view, which is why convex mirrors are used as car wing mirrors and shop security mirrors.

Limits of the model

The mirror equation assumes a small mirror compared with its radius of curvature, so that all rays stay close to the principal axis (the paraxial approximation). Real mirrors with a wide aperture suffer from spherical aberration, where rays striking the outer edge focus at a slightly different point, which is why precision instruments use parabolic rather than spherical mirrors. The equation also treats the mirror as forming a single sharp image, ignoring diffraction and the finite size of the light source.