What this calculator does
The lensmaker's equation gives the focal length of a lens from what it is physically made of and shaped like, rather than from measuring where it forms an image. It takes the refractive index of the glass, the curvature of each surface and the lens thickness.
The thin lens equation that most courses start with is this same equation with thickness set to zero. Including thickness matters for real lenses, especially strongly curved ones, and the calculator shows both so the difference is visible.
The formula
One over the focal length equals the refractive index minus one, multiplied by the difference in surface curvatures plus a correction term involving thickness. The optical power in dioptres is simply one over the focal length in metres.
| Term | Meaning |
|---|---|
| Refractive index (n) | How much the material slows light. Common crown glass is about 1.5. |
| Radius of curvature | The radius of the sphere each lens surface is part of. Sign conventions matter: positive when the surface is convex toward the incoming light. |
| Dioptre (D) | The reciprocal of focal length in metres. Spectacle prescriptions are written in these units. |
The inputs explained
| Field | What to enter |
|---|---|
| Refractive index of the lens | The refractive index of the lens material. |
| Radius of curvature R1 (+ if convex toward the light) (m) | Radius of curvature of the first surface, positive if convex toward the incoming light. |
| Radius of curvature R2 (− if convex away from the light) (m) | Radius of curvature of the second surface, negative if convex away from the incoming light. |
| Lens thickness (0 for thin-lens approximation) (m) | The lens thickness along its axis. Setting it to zero gives the thin lens approximation. |
When to use it
Designing a lens to a focal length
Choosing the curvatures and material to hit a target focal length is exactly what this equation is for.
Understanding a spectacle prescription
A prescription in dioptres corresponds to a specific focal length, and the lens shape follows from the material used.
Checking when thickness matters
Comparing the full result against the thin-lens approximation shows how much the thickness term contributes.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does refractive index change the focal length?
The same lens shape made from materials of increasing refractive index.
| Refractive index | Focal length | Optical power |
|---|---|---|
| 1.4 | 0.1268 m | 7.886 D (dioptres) |
| 1.5 | 0.1017 m | 9.833 D (dioptres) |
| 1.6 | 0.0849 m | 11.775 D (dioptres) |
| 1.8 | 0.0639 m | 15.644 D (dioptres) |
Questions
Why do the signs of the radii matter so much?
Because they encode which way each surface bulges. Getting a sign wrong turns a converging lens into a diverging one in the arithmetic, which is the most common error in applying the equation.
What does a negative focal length mean?
A diverging lens, which spreads light rather than bringing it to a focus. It forms virtual images and is used to correct short-sightedness.
Why are high-index lenses thinner?
Because a higher refractive index achieves the same bending with less surface curvature, and less curvature means less thickness at the centre or edge. The trade-off is usually more chromatic dispersion.
When can I ignore thickness?
When it is small compared with the radii of curvature, which covers most simple lenses. For thick or strongly curved lenses the correction term becomes significant and should be kept.
For where the image actually forms, see the thin lens equation calculator. For light bending at a single surface, see the Snell's law calculator.