Mirror Equation Calculator

What is the mirror equation?

The mirror equation is 1/f = 1/do + 1/di, where f = focal length, do = object distance from the mirror, and di = image distance from the mirror. Linear magnification M = −di/do = hi/ho. All distances are measured from the mirror's vertex along the principal axis.

What sign convention does this calculator use?

This calculator uses the real-is-positive convention: object distance do is always positive (a real object in front of the mirror); f is positive for concave mirrors and negative for convex mirrors; positive di means a real image forms in front of the mirror, while negative di means a virtual image behind the mirror. This convention is documented in the result panel of every calculation.

How are focal length (f) and radius of curvature (R) related?

For a spherical mirror, the focal length is half the radius of curvature: f = R/2. A concave mirror has positive R (center of curvature in front of the mirror), a convex mirror has negative R (center of curvature behind the mirror), and a plane mirror has infinite R.

How do I read real/virtual, upright/inverted, magnified/reduced images?

Look at the signs of di and M from the calculation: di > 0 means a real image (can be caught on a screen), di < 0 means a virtual image; M > 0 means an upright image, M < 0 means inverted; |M| > 1 means magnified, |M| < 1 means reduced, and |M| = 1 means the same size. The calculator displays all three classifications automatically as badges.

Give a worked example of the mirror equation.

Example: a concave mirror with f = 20 cm, object at do = 30 cm. Then 1/di = 1/f − 1/do = 1/20 − 1/30 = (3−2)/60 = 1/60, so di = 60 cm. Magnification M = −di/do = −60/30 = −2: the image is real (positive di), inverted (negative M), and 2 times larger than the object. You can verify this example directly in the calculator by selecting the di mode.

How does the mirror equation differ from the lensmaker's equation?

Both are about geometrical optics but the object differs: the lensmaker's equation (#1706 on this site) governs light refraction in LENSES (1/f = (n−1)(1/R1 − 1/R2)) with the Cartesian sign convention for two refracting surfaces; the mirror equation governs light reflection in MIRRORS (1/f = 1/do + 1/di) with the real-is-positive convention for a single reflecting surface. Don't mix the formulas: lenses use refractive index n, mirrors don't.

What kind of image does a convex mirror always produce?

A convex mirror (negative f) always produces a virtual, upright, and reduced image for any real object: di is always negative and |M| is always less than 1. That's why car side mirrors use convex mirrors — a wide field of view even though the image is small.

When does a mirror image form at infinity?

The image falls at infinity when the object is exactly at the focal point (do = f): the reflected rays exit parallel and never meet at a finite distance. The calculator flags this condition specially instead of showing a misleading number.

How does the mirror equation apply to plane mirrors?

A plane mirror is a special case with infinite f: the formula simplifies to di = −do and M = +1, meaning the image is always virtual, upright, and the same size, located as far behind the mirror as the object is in front. Select the Plane mirror type in the calculator for this mode.

Why can magnification (M) be negative?

The negative sign on M indicates orientation, not size: M = −2 means an inverted image twice as large, while M = +0.5 means an upright image half the object's size. Size is always read from the absolute value |M|. A common mistake is thinking that a negative M means a smaller image.