Radius of Curvature Calculator
Table of Contents
1. Radius of Curvature Tools
Solve R₁
Find the first surface radius when focal length, refractive index, and R₂ are known.
Solve R₂
Find the second surface radius when focal length, refractive index, and R₁ are known.
For the broad query radius of curvature calculator, most visitors are trying to recover one missing surface from the lens maker formula. Start with the common R₁ workflow below, then switch to R₂ if your second surface is unknown.
Radius of Curvature Calculator
Enter focal length (f), refractive index (n), and R₂ to calculate R₁
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
2. What Radius of Curvature Means
The radius of curvature is the radius of the imaginary sphere that matches a lens surface. A small radius means a strongly curved surface and more optical power. A large radius means a flatter surface.
In practice, a radius of curvature calculator is useful when you know the material and target focal length, but you still need to specify one surface during lens design or reverse engineering.
Smaller Radius
Stronger curvature, higher surface power, shorter focal length contribution.
Larger Radius
Flatter surface, weaker contribution, and approaches infinity for a plano face.
3. Formula and Sign Convention
Start with the thin-lens equation and rearrange it depending on which surface is unknown:
Solve R₁
R₁ = 1 / [1/(f(n-1)) + 1/R₂]
Solve R₂
R₂ = 1 / [1/R₁ - 1/(f(n-1))]
- Use positive R₁ for a typical first convex surface in a converging lens.
- Use negative R₂ for a typical second convex surface in a biconvex lens.
- Use `∞` for any plano surface because a flat surface has zero curvature.
4. When to Solve R1 vs R2
Use the R₁ calculator when
- You know the back surface already and need to shape the incoming face.
- You are iterating a front-surface machining or molding step.
- You are matching a known focal length with a chosen glass type.
Use the R₂ calculator when
- The first surface geometry is fixed by packaging or tooling.
- You are checking the exit-face curvature of an existing design.
- You want to compare two rear-surface options while holding R₁ constant.
5. Worked Examples
Example 1: Solve R₁ for a biconvex lens
Given: f = 0.10 m, n = 1.52, R₂ = -0.08 m
1 / [f(n-1)] = 1 / [0.10 × 0.52] = 19.23
1/R₂ = -12.50
1/R₁ = 19.23 + (-12.50) = 6.73
R₁ = 0.149 m
Example 2: Plano-convex design check
For a plano-convex lens, one radius is infinite. If the plano side is on the second surface, then R₂ = ∞ and the equation simplifies to R₁ = f(n-1). That shortcut is useful when comparing results from the plano-convex lens calculator.
6. FAQ
What is a radius of curvature calculator?
A radius of curvature calculator solves the missing lens surface radius from the lens maker formula using focal length, refractive index, and the other surface radius.
What is the difference between R1 and R2?
R₁ is the first surface encountered by incoming light. R₂ is the second surface. Their signs depend on the chosen optical sign convention.
Can I use a radius of curvature calculator for a plano surface?
Yes. A plano surface has infinite radius, so you treat that side as `∞` or zero curvature in the equation.
Which page should rank for the broad radius of curvature term?
Use this page for the broad query, then branch into the dedicated R₁ or R₂ calculators when you know which specific radius you need.
Related Tools & Guides
Radius R₁ Calculator
Dedicated calculator for the first surface radius.
Radius R₂ Calculator
Dedicated calculator for the second surface radius.
Radius of Curvature Formula
Learn the rearranged equations and sign rules.
Focal Length Calculator
Solve the lens focal length before working backward for curvature.
Plano-Convex Lens Calculator
A common case where one radius becomes infinite.
Biconvex Lens Focal Length
Check the symmetric two-surface case.