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.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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