Radius of Curvature Calculator: Lens & Mirror Optics

A radius of curvature calculator for optics works out the radius of a spherical lens or mirror surface from the focal length and the refractive index of the material. You enter the focal length, the refractive index of the lens material, and the surrounding medium (usually air), and the calculator returns the radius of curvature for each surface. For a simple thin lens with equal curvature on both sides, the result is a single radius value; for a mirror, the radius is simply twice the focal length.

Formula

The calculator uses the lensmaker’s equation for a thin lens with equal radii on both surfaces (a symmetric biconvex or biconcave lens):

1/f = (n – n₀) × (2 / R)

Solving for R gives:

R = 2 × f × (n – n₀) / n₀

• f = focal length of the lens (mm, cm, or m – keep consistent)
• n = refractive index of the lens material (unitless, e.g., 1.5 for common glass)
• n₀ = refractive index of the surrounding medium (1.0 for air, 1.33 for water)
• R = radius of curvature of each spherical surface (same unit as f)

For a mirror, the formula is simpler: R = 2f (the radius of curvature is twice the focal length, regardless of the mirror material).

Worked Example

Using the default values from the calculator above (f = 100 mm, n = 1.5, n₀ = 1.0):

1. Subtract the surrounding index from the lens index: 1.5 – 1.0 = 0.5
2. Multiply by 2: 2 × 0.5 = 1.0
3. Multiply by the focal length: 1.0 × 100 mm = 100 mm
4. Divide by the surrounding index (1.0) – no change: R = 100 mm

So a symmetric lens with a focal length of 100 mm and refractive index 1.5 in air has a radius of curvature of 100 mm on each surface. The optical power is 1000 / 100 = 10 diopters.

When to Use This Calculator

Use this tool when you are designing or selecting a spherical lens or mirror and need to know the physical curvature that produces a desired focal length. This is common in:

• Designing simple magnifying glasses or eyepieces
• Choosing a lens for a laser focusing setup
• Estimating the curve needed for a homemade telescope mirror
• Checking whether a stock lens will fit a given mount (curvature affects sag and edge thickness)

The calculator assumes a thin lens (negligible thickness) and equal radii on both sides. For real lenses with different front and back curves, you need the full lensmaker’s equation, which includes the lens thickness and both radii separately.

How to Read the Result

The radius of curvature is the distance from the vertex (center) of the lens surface to the center of curvature of that spherical surface. A larger radius means a flatter surface; a smaller radius means a more curved surface. For a positive focal length (converging lens), the radius is positive; for a negative focal length (diverging lens), the radius is negative. The sign convention matters when you specify the lens to a manufacturer.

In the example above, a 100 mm radius means the surface is part of a sphere with a 100 mm radius. If you were to grind a mirror with a 100 mm focal length, the radius would be 200 mm – a much shallower curve.

Common Mistakes and Unit Mix-Ups

• Mixing units – If you enter the focal length in mm, the radius comes out in mm. Don’t mix cm and mm in the same calculation.
• Forgetting the surrounding medium – The formula uses the index difference (n – n₀). If the lens is in water, the effective power changes dramatically. For example, a lens with n=1.5 in air (n₀=1) has a power factor of 0.5; in water (n₀=1.33) the factor is only 0.17, so the same curvature gives a much longer focal length.
• Using the mirror formula for a lens – For a mirror, R = 2f, but for a lens you must include the refractive index. A common error is to use the mirror formula for a lens, which gives a radius twice as large as it should be.
• Ignoring sign conventions – In optics, positive radii often indicate a surface that is convex to the incoming light. If you are specifying a lens, check the sign convention used by your supplier.

Limitations of the Thin-Lens Formula

This calculator uses the paraxial approximation, which means it is accurate for rays close to the optical axis and for lenses with small apertures. For wide-aperture lenses or thick lenses, spherical aberration becomes significant, and the actual focal length will differ from the thin-lens value. Also, the formula assumes the lens is symmetric (equal radii). Many real lenses are asymmetric to reduce aberrations; those require the full lensmaker’s equation:

1/f = (n – n₀) × (1/R₁ – 1/R₂ + (n – n₀) × d / (n × R₁ × R₂))

where R₁ and R₂ are the two surface radii and d is the lens thickness. For a thin lens (d ≈ 0) with R₁ = -R₂, this reduces to the formula used here.

Alternatives and Related Calculations

If you need to work with a thick lens or different front/back radii, use a full lensmaker’s equation calculator. If you are dealing with mirrors, the radius is simply twice the focal length, so no index is needed. For a plano-convex lens (one flat side, one curved), the formula becomes R = f × (n – n₀) / n₀ (half the value of the symmetric case). For a meniscus lens (both surfaces curve the same direction), the calculation is more complex and requires both radii.

You may also want to calculate the sagitta (depth of the curve) for a given radius and diameter, which is useful for grinding lenses. The sagitta s = R – √(R² – (d/2)²), where d is the lens diameter.

Frequently Asked Questions

What is the radius of curvature in optics?

It is the radius of the imaginary sphere that the lens or mirror surface is a part of. A flat surface has an infinite radius; a highly curved surface has a small radius. It determines how strongly the surface bends light.

How do I find the radius of curvature of a mirror?

For a spherical mirror, the radius of curvature is exactly twice the focal length: R = 2f. For example, a concave mirror with a focal length of 50 cm has a radius of 100 cm. This is independent of the mirror material.

What is the difference between focal length and radius of curvature?

Focal length is the distance from the lens or mirror to the point where parallel rays converge (or appear to diverge from). Radius of curvature is a geometric property of the surface itself. For a mirror, they are directly related (R = 2f); for a lens, the relationship depends on the refractive index.

Can I use this calculator for a convex lens?

Yes, if the lens is symmetric (both surfaces have the same radius). For a biconvex lens, the radius is positive; for a biconcave lens, it is negative. The calculator works for both, as long as you enter the correct focal length sign.

What if my lens has different front and back radii?

Then you need the full lensmaker’s equation, which includes both radii and the lens thickness. This calculator only handles the symmetric case. For asymmetric lenses, you can use the full formula or consult a lens design guide.

Does the surrounding medium matter?

Yes. The refractive index of the surrounding medium (air, water, oil) affects the lens power. A lens in water has a much longer focal length than the same lens in air because the index difference is smaller. The calculator includes the surrounding index as an input.

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in f | Focal Length | mm | 100 | 0
in n | Refractive Index | - | 1.5 | 1
in n0 | Surrounding Medium Index | - | 1 | 1
out R | Radius of Curvature (each surface) | mm | 2 * f * (n - n0) / n0 | 2
out P | Optical Power | diopters | 1000 / f | 2