Lens Maker’s Equation Calculator (focal length from radii and refractive index)

Understanding the lens maker's equation calculator — concept and principle

This calculator finds the focal length and optical power of a thin lens from the refractive index of its material, the index of the surrounding medium and the radii of its two surfaces. It also says whether the lens converges or diverges and shows the focal length of the same lens in air and in water. It is used in optics courses, lens design sketches and when making simple lenses.

The lens bends light only as much as its index exceeds the surroundings, so in water it focuses far weaker.

How it is calculated

1/f = (n_lens ÷ n_medium − 1) × (1/R₁ − 1/R₂)

Optical power = 1 ÷ f (in metres) = 100 ÷ f (in cm), in dioptres

Light enters through surface 1. A radius is positive if its centre of curvature lies on the outgoing side and negative if on the incoming side; enter 0 for a flat surface.
Air is taken as n = 1.000293 and water as 1.333.

The principle

Each curved surface bends light by an amount that depends on its curvature and on the jump in refractive index across it. A thin lens adds the bending of its two surfaces. The factor (n_lens/n_medium − 1) measures the index contrast: the bigger the contrast, the stronger the lens. That is why the same lens becomes much weaker in water, where the contrast is smaller. Thickness is ignored, which works well when the lens is much thinner than its radii.

Worked example

Default inputs: n_lens = 1.52, n_medium = 1.000293, R₁ = 10 cm, R₂ = −10 cm.

Index factor = 1.52 ÷ 1.000293 − 1 = 0.51955.
Curvature term = 1/10 − 1/(−10) = 0.2 cm⁻¹.
1/f = 0.51955 × 0.2 = 0.10391, so f = 9.62 cm, a converging (positive) lens.
Optical power = 100 ÷ 9.62 = 10.39 dioptres; same lens in air 9.62 cm.
In water: (1.52 ÷ 1.333 − 1) × 0.2 = 0.02806, so f = 35.64 cm.

Each point is the calculator re-run with only “Refractive index of the surrounding medium” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Refractive index of the surrounding medium	Focal length f (cm)
0.25	0.98
0.5	2.45
0.75	4.87
1	9.62
1.25	23.19
1.5	383.55
2	-20.81

How to read the result

A positive f means a converging lens that brings parallel light to a focus; a negative f means a diverging lens. Shorter focal length means a stronger lens. Swimmers see blurry underwater because the cornea's index contrast against water is small, much like the lens example here.

Common mistakes

Using the wrong sign convention for R₂ on a biconvex lens.
Forgetting the surrounding medium when the lens is not in air.
Applying the thin-lens formula to thick lenses.
Using an index measured at a different wavelength.

Keep in mind: The thin-lens equation ignores thickness and aberrations. Real optical designs use ray tracing.

Frequently asked questions

Why is a lens weaker in water?

The index difference between glass and water is much smaller than between glass and air.

What does a negative focal length mean?

The lens diverges light.

What is a dioptre?

The optical power of a lens with a 1 m focal length; 10 dioptres means f = 10 cm.

Can a lens be flat on one side?

Yes. Enter 0 for that radius; a plano-convex lens is common.

Sources: OpenStax, University Physics Volume 3 (2016), chapter 2 key equations (lens maker's equation)

Checked 9 Nov 2026. OpenStax, University Physics Volume 3 (2016), Chapter 2 Key Equations: lens maker’s equation 1/f = (n₂/n₁ − 1)(1/R₁ − 1/R₂), optical power P = 1/f; refractive indices from §1.1 Table 1.1.

Your numbers