Science calculator
Lensmaker's equation.
The focal length of a thin lens from its refractive index and the two surface radii. Instead of asking you to sign the radii, each surface is chosen as convex, flat or concave — the convention is applied for you, and the signed radii it used are shown so you can check.
- Optical power 1/f
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- Signed R₁ used
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- Signed R₂ used
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- n refractive index of the lens · R₁, R₂ the two surface radii (signed) · f focal length
Sign convention & assumptions
- Convex surface → positive radius. Concave → negative. Flat → infinite (adds no power). The same rule applies to both surfaces; the chooser sets the sign and shows the signed R₁ and R₂ it used above.
- Light enters at Surface 1 and exits at Surface 2, but the sign depends only on each surface's own shape, not the order.
- Worked shapes: biconvex (both convex) → converging; plano-convex (one flat); biconcave (both concave) → diverging; meniscus (one convex, one concave).
- Thin lens in air (thickness ≪ radii; surrounding index 1). A positive f converges, a negative f diverges.
How it works
Each surface bends light by an amount set by its curvature and the index step; the lensmaker's equation adds the two contributions. A more strongly curved surface (smaller radius) or a higher index gives a shorter focal length. Power in dioptres is simply 1/f with f in metres.
The convex/flat/concave chooser exists because the signs are where lensmaker's calculations usually go wrong. The rule here is uniform: a convex surface is a positive radius, a concave one negative, a flat one infinite. That pairs with the sum form 1/f = (n−1)(1/R₁ + 1/R₂) shown above — identical to the textbook 1/f = (n−1)(1/R₁ − 1/R₂) once R₂ is signed the Cartesian way. Thin-lens approximation only; a thick lens needs the extra centre-thickness term.
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