Lensmaker's Equation Calculator (Focal Length)

Where a lens gets its focal length

The thin lens calculator answers “where does the image form” once you know the focal length. The Lensmaker’s Equation answers the question before that: what focal length does this piece of glass actually have, given its shape and what it is made of? It is the equation lens designers and opticians use to grind a lens to a target focal length.

The thin-lens form

For a lens thin enough to ignore its thickness:

1/f = (n − 1) × (1/R1 − 1/R2)

Where:

• f = focal length (positive for converging, negative for diverging)
• n = refractive index of the lens material relative to its surroundings (about 1.5 for common crown glass, 1.33 if the lens sits in water)
• R1 = radius of curvature of the first surface (the one light hits first)
• R2 = radius of curvature of the second surface

The thick-lens correction

Real lenses have thickness. When the center thickness d is not negligible compared to the radii, the full form adds a third term:

1/f = (n − 1) × [ 1/R1 − 1/R2 + (n − 1) × d / (n × R1 × R2) ]

This calculator applies the thick-lens term whenever you enter a thickness, and drops it (thin-lens form) when you leave thickness blank.

The sign convention (this is where people go wrong)

Radii follow a sign rule based on which side the center of curvature sits on. The standard convention:

• A surface is positive if its center of curvature is on the outgoing (right) side of the lens, negative if on the incoming (left) side.
• For a typical biconvex converging lens: R1 is positive, R2 is negative.
• For a biconcave diverging lens: R1 is negative, R2 is positive.
• A flat surface has an infinite radius, so its 1/R term is zero. Leave that surface’s radius blank in this calculator to treat it as flat (useful for plano-convex and plano-concave lenses).

Getting a sign backwards flips a converging lens into a diverging one, so double-check against the lens shape you actually have.

Lens power in diopters

Opticians rarely talk about focal length directly. They use power:

P = 1 / f (with f in meters)

The unit is the diopter (D). A +2.00 D reading lens has a focal length of 0.5 m. A diverging lens for nearsightedness might be −3.00 D. The Lensmaker’s Equation is exactly how an optical lab turns a prescription in diopters into the curvature it grinds onto a blank.

Worked example, a biconvex crown-glass lens

Crown glass, n = 1.52. First surface R1 = +20 cm, second surface R2 = −20 cm (symmetric biconvex). Thin lens:

1/f = (1.52 − 1) × (1/20 − 1/(−20)) = 0.52 × (0.05 + 0.05) = 0.52 × 0.10 = 0.052 per cm

f = 1 / 0.052 = 19.2 cm. Positive, so it is converging, as expected for biconvex. Its power is 1 / 0.192 m = +5.2 D.

Why refractive index matters so much

The (n − 1) factor is the whole reason high-index lens materials exist. Switch the same shape from crown glass (n = 1.52) to a high-index plastic (n = 1.74) and the focal power jumps by (1.74 − 1) / (1.52 − 1) = 1.42, a 42 percent increase, with no change in curvature. That is why strong eyeglass prescriptions use high-index materials: the lens can be much thinner and flatter for the same correction. The chart below shows focal length shrinking as index rises for your entered curvature.

What the equation ignores

The Lensmaker’s Equation is a paraxial result, valid for rays close to the optical axis. It does not capture spherical aberration, chromatic aberration (n actually varies with wavelength, which is why prisms make rainbows), or coma. Real camera lenses stack many elements precisely to cancel these. But for a single lens and a first focal-length estimate, this equation is the workhorse, and it is what every optics course starts with.