Lensmaker's Equation Calculator
Find the focal length and optical power of any thin lens in air from its refractive index and the two radii of curvature of its surfaces.
cm
cm
Converging lens — positive focal length
- 1
n − 1 (index excess)
1.5 − 1 = 0.5 - 2
1/R₁ − 1/R₂ (curvature difference)
0.05 − (-0.05) = 0.1Flat surfaces (R = 0) contribute zero curvature. - 3
1/f = (n − 1) × (1/R₁ − 1/R₂)
0.5 × 0.1 = 0.05 - 4
Focal length f = 1 ÷ (1/f)
1 ÷ 0.05 = 20
How does this calculator work?
Lensmaker's equation: 1/f = (n−1)(1/R₁ − 1/R₂) where n > 1 is the glass index and R₁, R₂ are the signed surface radii (positive = centre of curvature to the right). Positive f = converging, negative f = diverging. Optical power in diopters = 100 / f_cm.
Formula
How this is calculated
The lensmaker's equation 1/f = (n − 1)(1/R₁ − 1/R₂) gives the focal length f of a thin spherical lens surrounded by air. Here n is the refractive index of the lens material and R₁, R₂ are the radii of curvature of the two surfaces. The formula uses the real-is-positive (Cartesian) sign convention: light travels left to right, R is positive if the surface's centre of curvature is to the right (first surface convex), negative if to the left. For a classic biconvex lens R₁ > 0 and R₂ < 0, which makes the factor (1/R₁ − 1/R₂) positive and, with n > 1, gives a positive focal length (converging lens). A flat surface contributes nothing (enter 0; its 1/R term is treated as zero).
Optical power P in diopters equals 1/f where f is in metres; entering radii in centimetres, the calculator converts automatically (P [D] = 100 / f [cm]).
This is the thin-lens approximation, valid when lens thickness is small compared with f and the radii. For thick lenses, add the thickness term t(n−1)²/(n R₁ R₂). The formula also assumes the surrounding medium is air (n_medium = 1); for an immersion lens replace (n−1) with (n_lens/n_medium − 1).
Frequently asked questions
Use the real-is-positive Cartesian convention (light left-to-right). R > 0 if the surface's centre of curvature is to the right; R < 0 if to the left. For a biconvex lens: R₁ > 0 (front convex), R₂ < 0 (rear also convex, curving away).
Enter 0 for that radius. The calculator treats 1/0 as zero (a flat surface has infinite radius and contributes zero curvature), matching the physical situation.
The refractive index of any real optical material is greater than 1 (air n ≈ 1, crown glass ≈ 1.52, sapphire ≈ 1.77). n ≤ 1 is unphysical for a standard lens in air.
Also known as
TG we-Calculate Editorial Team. (2026). Lensmaker's Equation Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/lensmakers-equation-calculator
TG we-Calculate Editorial Team. "Lensmaker's Equation Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/lensmakers-equation-calculator.
TG we-Calculate Editorial Team, "Lensmaker's Equation Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/lensmakers-equation-calculator
@misc{wecalculate_lensmakers_equation_calculator, title = {Lensmaker's Equation Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/lensmakers-equation-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?