Math & Science
Lens Power Calculator
Convert focal length to dioptres and back — and see which kind of vision the lens corrects.
Lens Power Calculator
About the Lens Power Calculator
Lens power is the reciprocal of focal length. Measure the focal length in metres, divide one by it, and the answer is in dioptres. This calculator converts in both directions — focal length to power, or power back to focal length — and accepts millimetres, centimetres, metres or inches.
It also tells you what kind of lens you are describing. A positive power means a converging (convex) lens, which corrects long-sightedness. A negative power means a diverging (concave) lens, which corrects short-sightedness.
This is the basic optics calculation, useful for physics coursework and understanding a prescription. It is not a surgical planning tool and does not compute intraocular lens power for cataract surgery, which needs biometry data and a validated clinical formula.
How to use the Lens Power Calculator
- Choose whether to solve for power or focal length.
- Enter the value you have, with the right unit.
- A negative focal length or power means a diverging lens.
- Optionally add a second lens to combine them.
- Select Calculate.
The formula
P = 1 ÷ f
P = power in dioptres (D), f = focal length in metres. Metres, not centimetres — this is where most errors start.
Worked example
A convex lens has a focal length of 25 cm. What is its power?
Convert to metres first: 25 cm = 0.25 m.
P = 1 ÷ 0.25 = +4.0 D
+4.0 dioptres — a converging lens
Using 25 instead of 0.25 gives 0.04 D, which is out by a factor of 100. Always convert to metres before dividing.
What a dioptre is
A dioptre measures how strongly a lens bends light. One dioptre is a lens that brings parallel light to a focus one metre away. Two dioptres focuses at half a metre, four dioptres at a quarter.
The useful property is that dioptres are linear in bending power while focal length is not. Going from 1 D to 2 D halves the focal length; going from 2 D to 3 D only cuts it by a third. Optometrists work in dioptres precisely because the numbers then behave predictably when lenses are combined.
| Power | Focal length | Type |
|---|---|---|
| +10.0 D | 10 cm | Strong converging |
| +4.0 D | 25 cm | Converging |
| +2.0 D | 50 cm | Typical reading glasses |
| +1.0 D | 1 m | Weak converging |
| −1.0 D | −1 m | Weak diverging |
| −2.5 D | −40 cm | Moderate short sight |
| −6.0 D | −16.7 cm | High short sight |
Positive and negative powers
- Positive (convex, converging) — thicker in the middle, bends light inward to a real focus. Corrects hyperopia (long-sightedness) and presbyopia, the age-related loss of near focus that sends most people to reading glasses in their forties.
- Negative (concave, diverging) — thinner in the middle, spreads light outward from a virtual focus. Corrects myopia (short-sightedness), where the eye focuses in front of the retina.
A negative focal length is not an error. It means the focus is virtual — light only appears to come from that point, and no image forms there on a screen.
Combining lenses
Place thin lenses close together and their powers simply add. This is another reason the dioptre is the convenient unit:
Ptotal = P₁ + P₂ + P₃ …
Valid for thin lenses in contact. Separate them and the spacing enters the calculation.
A +3.0 D lens with a −1.0 D lens gives +2.0 D overall. Adding focal lengths directly would give the wrong answer entirely — 33 cm plus −100 cm means nothing useful.
Reading a glasses prescription
The SPH (sphere) column is the lens power this calculator works with, in dioptres. But a prescription carries more:
- SPH — spherical power. Negative for short sight, positive for long sight.
- CYL — cylindrical power, correcting astigmatism. A different power in one meridian.
- AXIS — the orientation of that cylinder, 0 to 180 degrees.
- ADD — extra positive power in the lower part of a varifocal, for reading.
- PD — pupillary distance, used for centring rather than power.
Lens power is a reciprocal, so a small change in focal length moves it sharply at short distances. The roots calculator and percentage calculator are useful alongside it when working through optics problems by hand.
This tool converts the spherical power only. It cannot combine SPH and CYL into a single figure, and it is not a substitute for an eye test. Prescriptions come from an optometrist measuring your eyes, not from a calculation.
Other ways to use this calculator
- Dioptre calculator — optical power in dioptres is 1 ÷ focal length in metres.
- Convex concave lens — convex lenses have positive power, concave lenses negative.
Frequently asked questions
A dioptre is the unit of lens power, equal to the reciprocal of the focal length in metres. A lens that focuses parallel light one metre away has a power of 1 D; at half a metre it is 2 D. Dioptres are used instead of focal length because they add together simply when lenses are combined, and because they scale linearly with bending power.
It means the lens is diverging — concave, thinner in the middle — and spreads light outward instead of bringing it to a focus. Negative lenses correct myopia, or short-sightedness, where the eye focuses images in front of the retina. The focal length also comes out negative, which simply indicates a virtual focus: light appears to come from that point but no real image forms there.
For thin lenses held close together, add their powers directly: P total = P₁ + P₂. A +3.0 D lens combined with a −1.0 D lens gives +2.0 D. This only works in dioptres — adding focal lengths gives nonsense. If the lenses are separated by a meaningful distance, the spacing enters the calculation and the simple sum no longer applies.
The SPH (sphere) figure on your prescription is a lens power in dioptres, so yes for that column. But a prescription also carries CYL and AXIS for astigmatism, and ADD for varifocals, none of which this calculator combines. It converts spherical power to focal length and back. For an actual prescription, see an optometrist — it comes from measuring your eyes, not from arithmetic.
Sources and further reading
- BIPM — The International System of Units (SI Brochure) — the dioptre as reciprocal metre (m⁻¹)
- National Eye Institute — Refractive errors — myopia, hyperopia and how corrective lenses work
- Dioptre — the additive property of lens powers