Thin Lens Equation Calculator (Lens and Mirror Formula)
By Santaji GadepatilinUpdated October 27, 2026
Lenses and curved mirrors form images at distances set by their focal length. The thin lens equation predicts where the image appears and how large it is — the basis of cameras, glasses, microscopes and projectors.
How to use the thin lens equation calculator
Enter the focal length and object distance. The calculator finds the image distance, magnification and lens power. Use the sign convention below. Any two of f, dₒ and dᵢ determine the third.
The equations
1/f = 1/dₒ + 1/dᵢ · m = −dᵢ ÷ dₒ · P = 1 ÷ f
Sign convention (real-is-positive)
| Quantity | Positive | Negative |
|---|---|---|
| f | Converging lens, concave mirror | Diverging lens, convex mirror |
| dᵢ | Real image | Virtual image |
| m | Upright image | Inverted image |
Converging lens, f = 10 cm
| Object distance | Image distance | Magnification | Image |
|---|---|---|---|
| 50 cm | 12.5 cm | -0.25 | real, inverted, reduced |
| 30 cm | 15.0 cm | -0.50 | real, inverted, reduced |
| 20 cm | 20.0 cm | -1.00 | real, inverted, reduced |
| 15 cm | 30.0 cm | -2.00 | real, inverted, enlarged |
| 12 cm | 60.0 cm | -5.00 | real, inverted, enlarged |
| 8 cm | -40.0 cm | 5.00 | virtual, upright, enlarged |
| 5 cm | -10.0 cm | 2.00 | virtual, upright, enlarged |
- Object beyond 2f: real, inverted, smaller image (cameras, eyes).
- Object at 2f: same-size inverted image.
- Object between f and 2f: enlarged inverted image (projectors).
- Object inside f: virtual, upright, enlarged (magnifying glass).
Lenses in contact
Powers add: P = P₁ + P₂. Two +5 D lenses together make a +10 D lens with a 10 cm focal length.
Spectacles
Short-sighted (myopic) people wear diverging lenses with negative dioptres; long-sighted people and reading glasses use positive lenses.
Frequently asked questions
What is the thin lens equation?
1/f = 1/dₒ + 1/dᵢ, where f is the focal length, dₒ the object distance and dᵢ the image distance. The same formula works for curved mirrors.
What does a negative image distance mean?
The image is virtual — on the same side as the object for a lens — and cannot be projected on a screen. Magnifying glasses produce virtual images.
What does negative magnification mean?
The image is upside down (inverted). Positive magnification means upright.
What is a dioptre?
Lens power in reciprocal metres: P = 1 ÷ f. A +2.5 dioptre reading lens has a focal length of 40 cm.