Radius of Curvature Formula
Table of Contents
1. Core Formula
Start from the thin-lens lens maker equation:
Lens Maker Formula
1/f = (n-1)(1/R₁ - 1/R₂)
If focal length and refractive index are known, the radius of curvature formula lets you isolate the missing surface. That is exactly the workflow used in lens design, quality checks, and reverse engineering.
2. Solve for R1 or R2
Radius of Curvature Formula for R₁
R₁ = 1 / [1/(f(n-1)) + 1/R₂]
Use this version when the back surface is already known and you need to machine or inspect the first surface.
Radius of Curvature Formula for R₂
R₂ = 1 / [1/R₁ - 1/(f(n-1))]
Use this version when the front surface is fixed and you need to determine the second optical surface.
3. Formula Checker
Use the calculator below to verify the formula numerically. For the broad search term, use the dedicated radius of curvature calculator.
Radius of Curvature Formula Checker
Enter focal length (f), refractive index (n), and R₂ to verify the R₁ rearrangement
Formula
1/f = (n-1)(1/R₁ - 1/R₂)
Typical: 1.5 (glass), 1.33 (water), 1.52 (crown glass)
Positive for convex, negative for concave
Positive for converging, negative for diverging
4. Sign Convention and Units
- Keep all radii and focal length in the same unit system. Meters is the safest default.
- For a standard biconvex lens, R₁ is positive and R₂ is negative under the Cartesian sign convention.
- Plano surfaces use infinite radius, which removes that term from the equation.
- If the computed radius changes sign unexpectedly, re-check which surface is first and which direction light travels.
5. Worked Example
Given: f = 0.12 m, n = 1.50, R₂ = -0.10 m
1 / [f(n-1)] = 1 / (0.12 × 0.50) = 16.67
1/R₂ = -10.00
1/R₁ = 16.67 - 10.00 = 6.67
R₁ = 0.150 m
This is the same workflow used by the R₁ calculator. If the opposite side were unknown, you would switch to the R₂ rearrangement instead.
6. FAQ
What is the radius of curvature formula for a lens?
For lens design, it usually refers to solving R₁ or R₂ from the lens maker equation: 1/f = (n-1)(1/R₁ - 1/R₂).
Do I need a different formula for R1 and R2?
Yes. The base equation is the same, but the rearranged algebra is different depending on which surface is unknown.
Can I use the radius of curvature formula for plano-convex lenses?
Yes. The flat side has infinite radius, so that term becomes zero and the formula simplifies.
What causes wrong results most often?
Most mistakes come from mixing units, reversing the light direction, or using the wrong sign for R₁ or R₂.
Related Tools & Guides
Radius of Curvature Calculator
Check the formula with a working tool.
Radius R₁ Calculator
Solve the first surface directly.
Radius R₂ Calculator
Solve the second surface directly.
Focal Length Formula
Return to the parent equation before rearranging it.
Plano-Convex Lens Calculator
A common case where one radius is infinite.
Biconvex Lens Focal Length
See the symmetric two-curved-surface case.