Lens Equation Calculator
Solve 1/f = 1/dₒ + 1/dᵢ for focal length or distances, with magnification.
Magnification ≈ -0.5 (negative = inverted, real image)
The thin lens equation is 1/f = 1/dₒ + 1/dᵢ, and magnification is m = −dᵢ ÷ dₒ. For a converging lens with f = 10 cm and an object at 30 cm, the image forms at dᵢ = (10×30) ÷ (30−10) = 15 cm, with m = −0.5 — a real, inverted image at half size.
The thin lens equation
For a thin lens, 1/f = 1/dₒ + 1/dᵢ ties the focal length to the object distance (dₒ) and the image distance (dᵢ). Once you have those distances, the magnification m = −dᵢ/dₒ tells you how large the image is and whether it is upright or inverted.
f = focal length, dₒ = object distance, dᵢ = image distance, m = magnification
Worked example
A converging lens has f = 10 cm and an object at dₒ = 30 cm. Find the image distance and magnification:
- 1 Rearrange for the image distance. From 1/f = 1/dₒ + 1/dᵢ, isolate dᵢ = (f × dₒ) ÷ (dₒ − f).
- 2 Substitute the values. dᵢ = (10 × 30) ÷ (30 − 10) = 300 ÷ 20 = 15 cm — a real image past the lens.
- 3 Find the magnification. m = −dᵢ ÷ dₒ = −15 ÷ 30 = −0.5 — inverted and half the object’s size.
Sign conventions (real-is-positive)
The standard thin-lens convention used by this tool.
| Quantity | Positive (+) | Negative (−) |
|---|---|---|
| Focal length f | Converging (convex) | Diverging (concave) |
| Image distance dᵢ | Real image (other side) | Virtual image (same side) |
| Magnification m | Upright image | Inverted image |
| |m| value | > 1 enlarged | < 1 reduced |
Sign conventions and common mistakes
Focal length sets the lens type. A positive f is a converging (convex) lens; a negative f is diverging (concave). Get this sign wrong and every downstream result flips.
Magnification carries two pieces of information. Its sign tells orientation — negative means inverted (and, for a single lens, real) — while its magnitude tells the size: |m| > 1 is enlarged, |m| < 1 is reduced.
Common mistake: forgetting that the equation works in reciprocals. After solving for 1/dᵢ you must take the reciprocal again to get dᵢ itself — a step that is easy to skip.