Lensmaker's equation calculator
Enter the lens's refractive index, the radius of each surface with its sign and, for a thick lens, the thickness at the centre. The calculator applies the lensmaker's equation and shows the focal length, the power in dioptres and whether the lens converges or diverges light.
Runs in your browser: nothing you enter is sent anywhere.
How to use it
- Enter the refractive index of the lens material (about 1.5 for common glass and acrylic) and of the medium around it (1 for air).
- Enter both radii of curvature with their signs, 0 for a flat surface, and the centre thickness (0 for a thin lens).
- Read the focal length, the power and the lens type.
- Lensmaker’s equation: 1/f = (n_rel − 1) × [1/R₁ − 1/R₂ + (n_rel − 1) × d ÷ (n_rel × R₁ × R₂)], with n_rel = n ÷ n_medium.
- Thin lens (d = 0): 1/f = (n_rel − 1) × (1/R₁ − 1/R₂).
- Power: P = 1 ÷ f in dioptres (f in metres).
Worked example
A biconvex glass lens, n = 1.5, R₁ = 100 mm and R₂ = −100 mm, in air: 1/f = 0.5 × (1/100 + 1/100) = 0.01 per mm, so f = 100 mm and the power is 10 D.
The same lens 10 mm thick: 1/f = 0.5 × (0.02 − 0.5 × 10 ÷ (1.5 × 100 × 100)) = 0.5 × 0.0196667, so f = 101.6949 mm.
A plano-convex lens with R₁ = 50 mm and a flat back (R₂ = 0) also has f = 100 mm.
Tips
- Turning a lens around does not change its focal length.
- The focal length here is measured from the principal plane, which for a thick lens is not at the surface.
- Use the thin lens calculator to find where the image of an object forms once you know f.
Frequently asked questions
What is the lensmaker's equation?
For a thin lens in air, 1/f = (n − 1) × (1/R₁ − 1/R₂), where n is the refractive index and R₁, R₂ are the radii of curvature of the two surfaces. For a thick lens a third term adds (n − 1) × d ÷ (n × R₁ × R₂) inside the brackets.
Which sign do the radii take?
Light travels from left to right. A surface whose centre of curvature is to the right of it has a positive radius, one with the centre to the left a negative radius. So a biconvex lens has R₁ > 0 and R₂ < 0. Enter 0 for a flat surface.
What is optical power in dioptres?
Power P = 1 ÷ f with f in metres. A lens with f = 100 mm = 0.1 m has a power of 10 D; spectacle prescriptions use this unit, negative for diverging lenses.
Why does a lens focus less under water?
What matters is the index of the lens relative to its surroundings, n ÷ n_medium. Glass of n = 1.5 in water (1.333) bends light much less than in air, so the same biconvex lens with f = 100 mm in air has f ≈ 399 mm under water.