Mirror Equation Calculator
Solve for focal length, image or object distance using 1/f = 1/dₒ + 1/dᵢ for curved mirrors.
Sign convention: concave f > 0; convex f < 0; real image dᵢ > 0 (same side as object for mirrors)
What Is Mirror Equation?
The mirror equation 1/f = 1/dₒ + 1/dᵢ has the same form as the thin lens equation but with different sign conventions. For mirrors, the focal length f = R/2 where R is the radius of curvature. Concave mirrors have positive f; convex mirrors have negative f. Real images form in front of the mirror (same side as object), with positive dᵢ.
Concave mirrors can form both real and virtual images depending on object position. Beyond 2f: real, inverted, smaller image. At 2f: real, inverted, same size. Between f and 2f: real, inverted, magnified. At f: no image (parallel rays). Inside f: virtual, upright, magnified (like a makeup mirror). Convex mirrors always form virtual, upright, smaller images.
Magnification m = −dᵢ/dₒ. For mirrors, the sign convention gives m positive for upright images and negative for inverted images. |m| > 1 means enlarged; |m| < 1 means reduced. A makeup mirror (concave, object inside f) gives m > 1 and upright; a car side mirror (convex) gives m < 1 with a wide field of view.
Mirror applications span from cosmetics to astronomy. Concave mirrors focus parallel rays (satellite dishes, solar concentrators, telescope primaries, searchlights). Convex mirrors diverge rays and show wide-angle views (security mirrors, car door mirrors, the inside of spherical lamp globes). Parabolic mirrors avoid spherical aberration for point objects at infinity.
Formula Reference Table
| Solve For | Formula | Notes |
|---|---|---|
| Image distance | 1/dᵢ = 1/f − 1/dₒ | dᵢ > 0: real; dᵢ < 0: virtual |
| Object distance | 1/dₒ = 1/f − 1/dᵢ | dₒ > 0: real object |
| Focal length | 1/f = 1/dₒ + 1/dᵢ | f = R/2 (R = radius of curvature) |
| Magnification | m = −dᵢ/dₒ | −: inverted; + : upright |
| Concave | f > 0 | Can form real or virtual images |
| Convex | f < 0 | Only virtual, upright, smaller images |
3 Worked Examples
Face 15 cm from concave mirror, f = 25 cm.
- 1/dᵢ = 1/25 − 1/15 = 3/75 − 5/75 = −2/75
- dᵢ = −37.5 cm (virtual, behind mirror)
- m = −(−37.5)/15 = +2.5 (upright, 2.5× magnified)
Parallel starlight (dₒ → ∞) on concave f = 500 mm mirror.
- 1/dᵢ = 1/f − 1/∞ = 1/500
- dᵢ = 500 mm = f (light focuses at focal point)
- A small secondary mirror redirects light to the eyepiece
Object 3 m from convex mirror, f = −0.5 m.
- 1/dᵢ = 1/(−0.5) − 1/3 = −2 − 0.333 = −2.333
- dᵢ = −0.429 m (virtual, behind mirror)
- m = −(−0.429)/3 = +0.143 (upright, much smaller — wide field of view)
Real-World Applications
Common Mistakes to Avoid
Concave: f > 0 (center of curvature and focus are on the same side as the object). Convex: f < 0. Getting this wrong completely changes the result.
For mirrors, real images have positive dᵢ (on same side as object). For lenses, real images have positive dᵢ (on opposite side from object). The magnitude formula is the same but sign conventions differ.
dᵢ < 0 for mirrors means the image is virtual (behind the mirror, where reflected rays appear to diverge from). It cannot be projected on a screen.
The focal length of a spherical mirror is half the radius of curvature. A concave mirror with R = 20 cm has f = 10 cm.
m = −dᵢ/dₒ. For concave mirror with real image: dᵢ > 0, so m < 0 (inverted). For virtual image: dᵢ < 0, so m > 0 (upright).
Frequently Asked Questions
Related Physics Calculators
Formula Explorer connections
Interpretation: This relationship connects light propagation, geometry, wavelength, refraction, interference or image formation. Assumption: Use a consistent sign convention and units. Paraxial rays, thin elements, coherent light, vacuum wavelength or ideal optical components may be assumed.