Mirror Equation Calculator
Solve 1/f = 1/do + 1/di for the image distance formed by a curved mirror, plus magnification m = −di/do. Two 3D diagrams compare a concave mirror (real, inverted image) to a convex mirror (virtual, upright image), and charts show how image distance and magnification change as the object moves.
Reviewed by the ToolNestr Editorial Team — July 2026
Concave vs. convex mirrors
1. Concave mirror
Curves inward, converging reflected light to a real focal point in front of the mirror.
2. Convex mirror
Curves outward, spreading reflected light so it only appears to come from behind the mirror.
Mirror equation graphs
How it works
The core idea in one line: a curved mirror bends reflected light toward (concave) or away from (convex) a single focal point, and exactly where the object sits relative to that focal point determines whether the resulting image is real or virtual, upright or inverted, and bigger or smaller than the original.
1/f = 1/do + 1/di
mirror equation — f = focal length, do = object distance, di = image distance
m = −di/do
magnification — negative means inverted, positive means upright
hi = m × ho
image height from object height and magnification
Every spherical mirror has a focal length that describes how strongly it bends light — concave mirrors converge light to a real focal point in front of the mirror (positive f), while convex mirrors spread light apart so it only appears to come from a focal point behind the mirror (negative f). The mirror equation 1/f = 1/do + 1/di connects that focal length to any specific object distance, solving exactly where the image forms. The sign of the resulting image distance tells you everything: positive means the image is real and can be projected onto a screen; negative means it's virtual and only visible by looking directly into the mirror.
Worked example 1 — concave mirror, object at the center of curvature
Given: A concave mirror has f = 15 cm. An object sits do = 30 cm away (exactly at the center of curvature, R = 2f).
This is the special case where object and image distances are equal — it only happens when the object sits exactly at the center of curvature.
Worked example 2 — convex mirror (a car's passenger-side mirror)
Given: A convex mirror has f = −20 cm (negative, since it diverges light). An object sits do = 15 cm away.
This is why objects in a convex mirror look smaller and closer than they appear — and why these mirrors carry the familiar warning "objects in mirror are closer than they appear."
Image distance vs. object distance (concave mirror, f = 15 cm)
As the object moves closer to the focal point, the image distance grows dramatically — and eventually flips to virtual once the object moves inside f.
| Object distance (do) | Image distance (di) | Magnification | Image type |
|---|---|---|---|
| 45 cm (beyond 2f) | 22.5 cm | −0.50 | Real, inverted, reduced |
| 30 cm (at 2f) ★ | 30 cm | −1.00 | Real, inverted, same size |
| 20 cm (between f and 2f) | 60 cm | −3.00 | Real, inverted, enlarged |
| 10 cm (inside f) | −30 cm | +3.00 | Virtual, upright, enlarged |
★ Reference row (worked example 1). Crossing the focal point flips the image from real to virtual — this is the single most important qualitative shift to remember for concave mirrors.
Where the mirror equation actually matters
🚗 Vehicle safety mirrors
Convex mirrors are used for passenger-side mirrors and store security mirrors because their wide field of view (from the always-reduced image) lets a small mirror show a much larger area than a flat mirror could.
🔭 Reflecting telescopes
Large concave mirrors gather and focus starlight far more effectively than lenses of comparable size, since mirrors don't suffer from chromatic aberration — this is the working principle behind every major research telescope.
💇 Makeup and shaving mirrors
A concave mirror held closer than its focal length produces the enlarged, upright image used in magnifying vanity mirrors — move it past the focal point and the image flips and inverts.
🔦 Headlights and spotlights
Concave mirrors placed at the focal point of a light source reflect the light into a parallel beam, which is exactly why car headlights and flashlight reflectors are shaped as parabolic (near-spherical) concave mirrors.
Common misconceptions
"Concave mirrors always produce a magnified image."
Concave mirrors only magnify when the object is closer than the center of curvature (do < 2f). Beyond that distance, the image is real but smaller than the object — magnification depends entirely on where the object sits relative to f and 2f.
"A virtual image means no image actually forms."
A virtual image is a perfectly real, viewable image — you can see it by looking into the mirror. It's called virtual only because light rays don't actually converge there; they only appear to converge when traced backward, so the image can't be projected onto a screen the way a real image can.
"Convex mirrors can sometimes produce a real, inverted image."
For any real object, a convex mirror's negative focal length mathematically forces di to always be negative — the image is always virtual, upright, and reduced. This never changes regardless of how close or far the object is.
"The mirror equation only works for mirrors, not lenses."
The identical equation 1/f = 1/do + 1/di (with a different sign convention) also describes thin lenses — the underlying geometric optics is the same whether light is being reflected or refracted to form an image.
Formula sources & further reading
The formulas here are standard, traceable to:
- • OpenStax, University Physics Volume 3 — Chapter 2, "Geometric Optics and Image Formation" (free, peer-reviewed). openstax.org
- • Hecht, Optics — Spherical mirror imaging chapter.
- • Serway & Jewett, Physics for Scientists and Engineers — Geometric Optics.
1/f = 1/do + 1/di, m = −di/do. Sign convention: concave f > 0, convex f < 0; real image di > 0, virtual image di < 0. Results are rounded for display.
How to use this calculator
Enter focal length with correct sign
Positive for concave (converging), negative for convex (diverging) mirrors.
Enter object distance
Always a positive value for a real object placed in front of the mirror.
Read image distance and magnification
Positive di = real image; negative di = virtual image. The sign of m tells you upright vs. inverted.
Related tools
Frequently asked questions
What is the mirror equation?
The mirror equation is 1/f = 1/do + 1/di, where f is the focal length, do is the object distance from the mirror, and di is the image distance. It applies to both concave and convex mirrors using a sign convention.
What is the sign convention for mirrors?
Concave (converging) mirrors have a positive focal length; convex (diverging) mirrors have a negative focal length. A positive image distance (di) means a real image in front of the mirror; a negative di means a virtual image behind the mirror.
How is magnification calculated?
Magnification is m = −di/do. A negative m means the image is inverted; a positive m means upright. |m| > 1 means enlarged, |m| < 1 means reduced.
Why do convex mirrors always produce a smaller, upright image?
A convex mirror has a negative focal length, which forces the mirror equation to always produce a negative (virtual) image distance for any positive object distance — the image is always virtual, upright, and reduced, which is exactly why convex mirrors are used for wide-view safety mirrors.
What happens when the object is exactly at the focal point of a concave mirror?
When do = f, the mirror equation gives 1/di = 1/f − 1/f = 0, so di is infinite — the reflected rays emerge perfectly parallel and never converge to form an image at any finite distance.