Mirror Equation Calculator
Calculate image distance, magnification, and image properties for curved mirrors using 1/f = 1/d_o + 1/d_i.
Works for concave and convex mirrors.
Mirror Equation
For a curved mirror with focal length f, an object at distance d_o produces an image at distance d_i where:
1/f = 1/d_o + 1/d_i
The magnification (size ratio) is:
m = −d_i / d_o
Sign Convention (Cartesian / “real-is-positive”)
| Quantity | Sign |
|---|---|
| Concave mirror f | Positive |
| Convex mirror f | Negative |
| d_o (object in front) | Positive |
| d_i (real image, same side as object) | Positive |
| d_i (virtual image, behind mirror) | Negative |
| m > 0 | Upright |
| m < 0 | Inverted |
| m | |
| m |
The radius of curvature R is related by R = 2f.
Image Type Quick Reference
| Mirror | Object Position | Image |
|---|---|---|
| Concave | Beyond C (d_o > 2f) | Real, inverted, reduced |
| Concave | At C | Real, inverted, same size |
| Concave | Between C and F | Real, inverted, enlarged |
| Concave | At F | At infinity (parallel rays) |
| Concave | Inside F | Virtual, upright, enlarged |
| Convex | Anywhere | Virtual, upright, reduced |
| Plane | Anywhere | Virtual, upright, same size |
Worked Example — Concave Mirror, f = 10 cm, d_o = 30 cm
- 1/d_i = 1/10 − 1/30 = 3/30 − 1/30 = 2/30
- d_i = 15 cm (positive → real image)
- m = −15/30 = −0.5 (inverted, half-size)
So the image forms 15 cm in front of the mirror, upside down, half the object’s size — exactly the kind of image a shaving mirror produces when held far from your face.
Worked Example — Convex Car Side Mirror
Convex mirrors always produce virtual, upright, reduced images. That is why side mirrors are stamped “objects in mirror are closer than they appear” — the image distance is smaller than the actual object distance.
Limitations
The thin-mirror equation assumes paraxial rays — rays close to the optical axis. For wide-aperture or high-speed optics, spherical aberration, coma, and astigmatism produce real-world deviations from the simple formula. Telescopes use parabolic instead of spherical mirrors specifically to eliminate spherical aberration on-axis.