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Lensmaker's Equation Calculator

Find a lens's focal length from its curvature.

Result

20.00 cm
5.00 D

How it is calculated

The tool evaluates 1/f = (n − 1) × (1/R₁ − 1/R₂) and then takes f = 1 / (1/f). n is the refractive index of the lens material (must be greater than 1), R₁ and R₂ are the radii of curvature of the first and second surface in centimetres, and f is the focal length in centimetres. Optical power in dioptres is P = 100 / f with f in cm (equivalently 1 / f in metres).

Signs matter: a surface bulging toward the incoming light has a positive radius. A positive f means a converging lens, a negative f a diverging one. The tool returns nothing if n ≤ 1, if either radius is 0, or if the two terms cancel (1/R₁ = 1/R₂, a zero-power lens). It uses the thin-lens form and ignores lens thickness.

Worked example

Take a symmetric biconvex lens with n = 1.5, R₁ = 20 cm and R₂ = −20 cm. First 1/R₁ − 1/R₂ = 0.05 − (−0.05) = 0.10. Then 1/f = (1.5 − 1) × 0.10 = 0.05 per cm, so f = 1 / 0.05 = 20 cm. The power is 100 / 20 = 5 D. The tool shows a focal length of 20.00 cm and a power of 5.00 D.

Try this example →

Common mistakes

  • Giving both radii the same sign for a biconvex lens. The second surface curves the other way, so it needs the opposite sign (20 and −20, not 20 and 20, which gives no result).
  • Entering radii in millimetres or metres. The focal length is reported in cm, so radii should be in cm too.
  • Typing the refractive index as 1 or lower. Air has n = 1, and a lens needs a value above it, such as 1.5 for typical glass.
  • Ignoring a negative answer. A negative focal length and negative dioptres mean the lens diverges light, as in a lens for short-sightedness.
  • Using the result for a thick lens or a lens in water. The formula assumes a thin lens surrounded by air.

About this calculator

This calculator applies the lensmaker's equation, 1/f = (n−1)(1/R₁−1/R₂), which comes from the thin-lens approximation of Snell's law at each curved surface. The refractive index n describes how strongly the glass or plastic bends light compared to air, while R₁ and R₂ are the signed radii of curvature of the front and back surfaces; together they determine the focal length f and, as its reciprocal, the optical power in diopters.

Lens designers use this relationship to shape eyeglass lenses, camera and telescope objectives, and microscope optics before ever grinding glass, since it predicts focal length directly from geometry and material. Optical engineers, opticians fitting prescriptions, photography and telescope hobbyists, and physics students studying geometric optics all rely on it to size a lens for a target focal length or check how a given curvature and material will perform.

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Comments (1)

  • Editorial team

    A flat face has an infinite radius, but the tool rejects a radius of 0, so for a plano-convex lens enter a very large number such as 100000 for the flat side. With n = 1.5 and R1 = 20 cm that returns about 40 cm, matching the rule that this kind of lens has f equal to twice its curved radius. A symmetric biconvex lens of the same index has f equal to its radius. Which lens shape are you designing?

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