The lens maker's equation connects a lens's physical shape — its material and the curvature of its two surfaces — to the optical behavior that shape produces: focal length and lens power. This calculator computes both directly from n, R1, and R2, and can also work backward to solve for any one of the four quantities (n, R1, R2, or f) given the other three.

How the Lens Maker's Equation Calculator works

For a thin lens (one whose thickness is small compared to R1 and R2), the focal length is given by 1/f = (n − 1)(1/R1 − 1/R2). The (n − 1) term captures how much more the lens material bends light than the surrounding air; the (1/R1 − 1/R2) term captures how much the two surfaces' curvatures combine to focus or spread that bent light. Once f is known, lens power follows directly as P = 1/f (in diopters, with f measured in meters), and the sign of f classifies the lens as converging (f > 0) or diverging (f < 0).

Inputs and what they mean

Refractive index (n) is a unitless property of the lens material — ordinary glass is roughly 1.5, and higher-index materials let you achieve the same power with less curvature (thinner, flatter lenses). R1 and R2 are the radii of curvature of the first and second surfaces the light hits, sharing whatever length unit you enter them in (centimetres here); their signs follow the convention that a radius is positive when its center of curvature lies on the far side of the lens from the incoming light, negative when it lies on the near side. Getting the sign wrong is the single most common mistake with this formula — a biconvex lens is R1 > 0, R2 < 0, and simply reversing those signs turns the same magnitudes into a biconcave, diverging lens instead.

Limits and edge cases

This calculator uses the thin-lens approximation, which ignores the lens's physical thickness and any spherical aberration from real, non-ideal surfaces — it is accurate for typical corrective and camera lenses but not for thick or highly curved specialty optics. A flat surface has an infinite radius of curvature rather than zero, so a plano-convex lens is modeled by making one radius very large rather than entering zero (entering 0 for R1 or R2 is treated as invalid, since it implies a surface curved to a single point). If R1 equals R2 exactly, the two curvature terms cancel to zero optical power — a flat piece of glass — which this calculator flags rather than returning an undefined infinite focal length. The lens maker's equation only outputs the focal length and power of the lens itself; to find where an image actually forms from a given focal length and object distance, see the Lens Equation Calculator.