Last updated: 2026-06-15

The Lens-Maker's Equation

Where the thin-lens equation tells you where an image forms, the lens-maker's equation tells you what a lens is — it predicts the focal length from the glass and the grinding alone. Give it the refractive index of the material and the curvature of the two faces, and it returns the focal length and optical power. This calculator does exactly that.

The formula

For a thin lens in air,

where $n$ is the refractive index of the lens material, and $R_1$ and $R_2$ are the radii of curvature of the front and back surfaces. The optical power is just the reciprocal of the focal length,

measured in dioptres (D), where one dioptre is one reciprocal metre.

The sign convention

Curvature carries a sign. A surface that bulges toward the incoming light has a positive radius; one that caves away has a negative radius. So a symmetric biconvex lens has $R_1 > 0$ and $R_2 < 0$, which makes the bracket large and positive — a converging lens. Flip both signs for a biconcave lens and the focal length turns negative, marking a diverging lens. A flat face has infinite radius, so its $\tfrac{1}{R}$ term is simply zero.

Worked example

A symmetric biconvex lens of crown glass, $n \approx 1.5$, with both surfaces ground to a radius of $10\ \text{cm} = 0.1\ \text{m}$, so $R_1 = +0.1\ \text{m}$ and $R_2 = -0.1\ \text{m}$:

so the focal length is

How it differs from the thin-lens equation

The two equations work in sequence. The lens-maker's equation is a property of the lens: feed in shape and material once, and you have the focal length. The thin-lens equation, $1/f = 1/d_o + 1/d_i$, then takes that focal length and locates the image for any object distance. Characterise the lens first, then image with it as often as you like.