The lens maker equation calculator above works out what a piece of glass will do to light before the lens exists. Give it the refractive index of the material and the radius of curvature of each surface, and it returns the focal length, the optical power in dioptres and a plain statement of whether the result converges or diverges. It handles a flat surface, a surrounding medium other than air, and the centre-thickness correction that separates a real lens from the textbook idealisation.
Arb Digital builds free calculators that stop at a clear boundary. This page starts from the glass and produces a focal length. The site's thin lens equation calculator starts from a focal length you already have and produces an image position and magnification. Run this page first if you are designing or identifying the lens, and that one afterwards if you are working out where the image lands.
What This Lens Maker Equation Calculator Does
In its default mode it computes focal length from R₁, R₂, the lens index and the surrounding medium's index. Switch modes and it will instead find the refractive index that a measured focal length implies — which is how you identify unknown glass on a bench — or the second surface radius needed to reach a target focal length when the first surface is already fixed by the tooling you have.
The shape presets exist because the sign convention is where almost everyone slips. Choosing Equal biconvex sets R₁ positive and R₂ negative with the same magnitude; Plano-convex makes the second surface flat; Positive meniscus makes both radii the same sign, which is the shape almost every spectacle lens actually uses. The presets take the magnitude currently in the R₁ field, so set that first and then pick a shape.
The thick-lens option adds the term that depends on centre thickness. It changes the answer very little for a shallow lens and a great deal for a strongly curved one, and the results grid always shows both the thin-lens value and the difference so you can see whether it was worth including. That comparison is the reason both models are on one page rather than two.
How to Use It
- Fix the sign convention in your head first. Light travels left to right; a radius is positive when its centre of curvature sits on the outgoing side of the surface.
- Enter the radii in millimetres, with signs. A flat surface has an infinite radius, so type 0 and the tool treats it as flat rather than dividing by zero.
- Set the two refractive indices. The lens index does the work; the medium index is 1 in air and 1.33 in water, and the difference between them is what bends light.
- Choose thin or thick. Start thin. If the grid shows a meaningful shift when you switch to thick, your lens is thick enough that the correction matters.
- Read the dioptre figure if you are working in optics. It is simply 1,000 divided by the focal length in millimetres, and it is what prescriptions and catalogue listings use.
The Formula: How the Lens Maker Equation Works
For a thin lens the relation is 1/f = (n₂/n₁ − 1)(1/R₁ − 1/R₂), where n₂ is the index of the lens, n₁ the index of the surrounding medium, and R₁ and R₂ the radii of the first and second surfaces. OpenStax University Physics Volume 3, section 2.4 on thin lenses, gives it in exactly that form and reduces it to 1/f = (n − 1)(1/R₁ − 1/R₂) for a lens in air.
Work the default values. Crown glass at n = 1.52 in air, formed into an equal biconvex lens with R₁ = +100 mm and R₂ = −100 mm, gives 1/f = 0.52 × (1/100 − 1/−100) = 0.52 × 0.02 = 0.0104 per millimetre. The focal length is therefore 96.15 mm, and the optical power is 1,000 ÷ 96.15 = 10.4 dioptres.
The thick-lens form adds one term: 1/f = (n − 1)[1/R₁ − 1/R₂ + (n − 1)d ÷ (nR₁R₂)], with d the centre thickness. For the same lens with d = 5 mm the extra term is 0.52 × 5 ÷ (1.52 × 100 × −100) = −0.000171, so 1/f becomes 0.52 × 0.019829 = 0.010311 and the focal length lengthens to 96.98 mm. Less than one per cent here, and much larger on a short-focus lens.
The Sign Convention Is the Whole Difficulty
Both radii are measured from the surface to its centre of curvature, along the direction light is travelling. A radius counts as positive when that centre lies on the outgoing side, and negative when it lies on the incoming side. For a biconvex lens the first surface bulges toward the incoming light, so its centre is downstream and R₁ is positive; the second surface bulges away, its centre is upstream, and R₂ is negative.
This is why an equal biconvex lens has (1/R₁ − 1/R₂) equal to 2/R rather than zero. Enter both radii as positive and the bracket collapses to zero, the focal length runs to infinity, and the tool will tell you the lens has no focusing power. That is not a bug in the equation — it is the correct answer for a lens whose surfaces are parallel curves, which is what two identical same-sign radii actually describe.
A meniscus lens genuinely has two same-sign radii, and it still focuses, because the magnitudes differ. A positive meniscus has a steeper front surface than back; a negative meniscus is the other way round. Almost every spectacle lens is a meniscus, chosen so the concave rear surface wraps toward the eye and keeps the visual field usable off-axis. Try the spectacle preset and then flip the sign of one radius to see how completely the answer changes.
What Changes When the Lens Is Not in Air
The bending power of a surface depends on the difference between the two indices, not on the lens index alone. A crown glass lens at n = 1.52 has a factor of 0.52 in air. Put the same lens in water at n = 1.33 and the factor becomes 1.52/1.33 − 1 = 0.143, so its focal length nearly quadruples and its power drops by the same ratio. This is exactly why human vision is so blurred underwater: the cornea does most of the eye's focusing precisely because it borders air, and immersing it removes most of that power.
Push the medium index above the lens index and the factor goes negative, which inverts the lens entirely — a converging shape becomes a diverging element. Air bubbles in water behave this way, which is why a bubble acts as a diverging lens despite looking like a fat convex one. The OpenStax section on refraction sets out the index values behind this, including the 1.33 figure for water derived from measured refraction angles. The site's index of refraction calculator handles that side of the problem directly.
Focal Length, Power and Why Opticians Use Dioptres
Optical power in dioptres is the reciprocal of focal length in metres, so a 100 mm lens is 10 dioptres and a 500 mm lens is 2. The reason the trade works in power rather than focal length is that powers of thin lenses in contact simply add, whereas focal lengths do not. Stack a +3 and a +2 and you have a +5; stacking a 333 mm and a 500 mm lens gives 200 mm, which is far less obvious.
Negative powers describe diverging lenses, where the focal length is negative and parallel light spreads out as if from a virtual point in front of the lens. The tool labels this in the grid rather than leaving you to interpret a minus sign. Our diopter calculator converts between the two conventions and handles lens stacks; this page is where the dioptre figure comes from in the first place.
Where This Equation Stops Being Accurate
The lens maker equation is a paraxial result. It assumes rays close to the optical axis and small angles, so that the sine of an angle can be replaced by the angle itself. Real lenses used at wide aperture depart from it through spherical aberration — marginal rays focus closer than paraxial ones — and the equation gives no warning that this is happening.
It also assumes a single wavelength. Refractive index varies with colour, so the focal length you calculate is only correct for the wavelength whose index you entered. Crown glass around n = 1.52 is normally quoted at the yellow sodium line; blue light sees a slightly higher index and focuses closer. That spread is chromatic aberration, and it is the reason quality lenses are cemented pairs of different glasses rather than single elements.
Finally, the thick-lens result gives the effective focal length measured from the lens's principal planes, not from either glass surface. For a thick or a compound lens the back focal distance — the physical gap from the rear surface to the focus — is shorter than the effective focal length, and mounting hardware has to be designed against the back focal distance rather than this number.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Entering both radii positive for a biconvex lens — the second surface is negative under the standard convention, and getting it wrong makes the focal length infinite or the wrong sign.
- Treating a flat surface as radius zero in the arithmetic — a plane has infinite radius, which contributes nothing to the bracket. Type 0 here and the tool applies that correctly for you.
- Using the lens index alone when the lens is not in air — it is the ratio of lens index to medium index that matters, and immersion can gut a lens's power.
- Ignoring wavelength — the index you enter fixes the colour the answer applies to, and blue light will focus measurably closer than red.
- Measuring the focal length from the glass surface — the thick-lens result is measured from the principal planes, and the back focal distance is a different and shorter number.
Related Free Tools From Arb Digital
Take the focal length from this page into the thin lens equation calculator to find image distance and magnification, or into the diopter calculator to work in optical power. Curved mirrors follow a related but distinct relation on the mirror equation calculator, and the material side is handled by the index of refraction calculator. Photographers will want the depth of field calculator once a focal length is settled, and polarising elements in the same optical train are covered by the Malus law calculator. Rescale dimensions with the length converter, and browse everything on the free online tools hub.
Frequently Asked Questions
Light is taken to travel from left to right, and a surface radius is positive when its centre of curvature lies on the outgoing side of that surface. Under this convention an equal biconvex lens has a positive first radius and a negative second radius.
Type 0 in the radius field. A plane surface has an infinite radius of curvature, and the tool converts your 0 into that infinity so the surface contributes nothing to the focusing term rather than causing a division by zero.
When the centre thickness is a meaningful fraction of the radii. The grid shows the thin-lens answer and the shift the thickness term produces, so switch the model over and look at that shift. If it is negligible for your lens, stay with the thin form.
Because the focusing factor is the ratio of the lens index to the surrounding medium's index, minus one. In air a lens at index 1.52 has a factor of 0.52; in water at 1.33 that factor drops to about 0.14, so the focal length lengthens by roughly the same ratio.
This page derives focal length from the physical lens — its glass index and its two surface radii. The thin lens equation takes a focal length as given and relates object distance to image distance. One tells you what lens you have; the other tells you where it puts the image.
Yes, through the refractive index. Index rises toward the blue end of the spectrum, so blue light focuses slightly closer than red for the same lens. The focal length reported here applies only to the wavelength whose index you entered.
Because the bracket in the equation has come out as zero, which happens when the two surfaces have identical radii of the same sign, or when the lens and medium indices are equal. Both describe a shape that bends parallel light not at all, so the focal length is infinite.
This tool is provided for educational and estimating use. It applies the paraxial lens maker equation at a single wavelength and does not model aberrations, coatings, apertures or compound lens assemblies, so treat its output as a first-order optical result rather than a lens design.
