A lens corrects vision by putting light into a particular vergence at the eye. Move the same lens closer to or further from the cornea and the vergence arriving at the eye changes, so the power needed to produce the identical result changes too. That is the whole of vertex distance: a spectacle lens sits roughly twelve millimetres in front of the cornea, a contact lens sits on it, and the published vertex formula converts between the two planes.
Arb Digital publishes free physics calculators that show their working and state their limits. This one is a teaching tool for geometrical optics. It answers one narrow question — what power at the corneal plane produces the same vergence as a given power at the spectacle plane — and it does not produce a contact lens prescription, for reasons set out in full below.
What This Contact Lens Vertex Calculator Does
You enter a spectacle sphere, an optional cylinder, and the vertex distance in millimetres. The tool converts each principal meridian separately, reports the sphere at the corneal plane, rebuilds the cylinder from the difference between the two converted meridians, and shows both the raw arithmetic and the value rounded to the step lenses are actually made in.
The direction is reversible. Converting from the corneal plane back to the spectacle plane uses the same relation with the sign of the distance flipped, which is useful for checking that a conversion you were given is self-consistent.
The most useful output is arguably not the converted power at all but the change in power, shown in the grid. Below about four dioptres that change is smaller than the step lenses are made in, so the conversion changes nothing you could order. Above it, the change grows quickly. Seeing where that threshold sits is the point of the page.
How to Use It
- Enter the spectacle power as written. Sphere first, cylinder second, keeping the sign convention your prescription uses. Mixing plus-cylinder and minus-cylinder notation between the two boxes will give a meaningless second meridian.
- Use a measured vertex distance if you have one. Twelve millimetres is a textbook default, not a fact about your frames. A frame that sits close to the face and one that sits well forward can differ by several millimetres, and at high powers that difference is visible in the answer.
- Read the change in power, not only the converted power. If the change is smaller than a quarter of a dioptre, the conversion has made no practical difference at all.
- Compare the rounded and unrounded values. Rounding is where a conversion either does or does not survive contact with what is actually manufactured.
- Take the result to an eye-care professional. This page has produced a number in dioptres. A contact lens prescription is a different and much larger object, and the difference is explained below.
The Formula: How Vertex Conversion Is Calculated
Write the spectacle power as Fs in dioptres and the vertex distance as d in metres. The power at the corneal plane is
Fc = Fs ÷ (1 − d × Fs)
The formula follows directly from vergence arithmetic. A lens of power Fs produces light converging to, or diverging from, a point at distance 1/Fs from the lens. Shift the reference plane by d and that distance becomes 1/Fs − d, whose reciprocal is the power required at the new plane. Rearranging gives the expression above. The same relation runs backwards with d replaced by −d, which is what the direction selector does.
Work the default. A spectacle sphere of −8.00 D at a vertex distance of 12 mm gives d × Fs = 0.012 × (−8) = −0.096, so the denominator is 1.096 and the corneal-plane power is −8 ÷ 1.096 = −7.299 D. Rounded to the nearest quarter dioptre that is −7.25 D, a change of 0.75 D from the spectacle value.
Now the same distance with a plus lens. A spectacle sphere of +8.00 D gives a denominator of 1 − 0.096 = 0.904 and a corneal-plane power of +8.85 D, rounding to +8.75 D. Note that the two move in opposite directions: minus powers get weaker at the eye, plus powers get stronger. That asymmetry catches people out, and it is a consequence of the sign of d × Fs in the denominator rather than of anything about the eye.
The vergence reasoning underneath this is the same thin-lens arithmetic set out in the Georgia State University HyperPhysics page on the thin lens equation, and the relationship between focal length and dioptric power is handled separately by the diopter calculator on this site, which converts a focal length to a power but deliberately does no vertex work.
Why the Conversion Mainly Matters Beyond About Four Dioptres
This is the single most misunderstood thing about vertex conversion, and it is worth stating plainly rather than implying that every prescription needs adjusting.
At 12 mm, the quantity d × Fs is small for low powers. At −2.00 D it is 0.024, and the converted power is −1.95 D — a change of 0.05 D, which is a fifth of the smallest step a lens is made in. At −4.00 D the change is about 0.18 D, still under a quarter dioptre, though close enough to it that rounding starts to matter. At −6.00 D it is around 0.40 D and unambiguously real; at −10.00 D it is over a dioptre.
So the working rule that appears in optical training is that vertex conversion becomes clinically relevant somewhere around four dioptres, and below that it is arithmetic without consequence. The threshold is not a sharp line and it moves with the vertex distance: a frame sitting at 16 mm reaches the same change at a lower power than one sitting at 10 mm. Set the vertex distance box to different values and watch where the change in power crosses a quarter of a dioptre — that is the honest version of the rule.
A Contact Lens Prescription Is Not a Spectacle Prescription
This is the part that matters most, and no calculator can do it. Converting a power is one line of arithmetic. Prescribing a contact lens is not.
A contact lens prescription specifies, at minimum, the power, the base curve that matches the curvature of your cornea, the diameter that gives adequate coverage and movement, the material and its oxygen transmissibility, the wearing schedule and replacement interval, and the specific brand and product it was fitted in — because the same nominal power in two different products does not sit the same way on the eye. None of those follow from a spectacle prescription, and none of them can be derived from a number typed into a web page.
Fitting is a physical assessment. A practitioner measures corneal curvature, evaluates the tear film, checks how the lens centres and how much it moves on blink, examines the cornea for the changes that a poorly fitting or insufficiently oxygen-permeable lens produces, and reassesses after a period of wear. A lens that sits too tight restricts tear exchange; one that sits too flat moves excessively and decentres. Neither problem is visible in the power. Contact lenses are regulated as medical devices in most jurisdictions precisely because a badly fitted one can damage the cornea, and the patient-facing explanation of what a fitting involves is set out by the American Academy of Ophthalmology's Contact Lens 101 guide.
There are also optical effects the vertex formula ignores. Moving a correction onto the eye changes retinal image size and changes the accommodative and convergence demand, which is why the near correction for a presbyopic contact lens wearer is not simply the spectacle addition converted. Higher astigmatic corrections may be better addressed by a toric soft lens, a rigid lens or a different design entirely rather than by the converted cylinder this page computes. Those are clinical judgements.
Where the Vertex Distance Number Actually Comes From
Twelve millimetres is the value most commonly assumed, and many published conversion tables are built on it. It is a reasonable average for a conventional frame, but it is an average. The distance depends on the frame's shape, how far down the nose it sits, the pantoscopic tilt, and the wearer's facial anatomy. It is measured in practice with a distometer or from a fitting cross reference, and a prescription written for high-power lenses will often record the vertex distance it was refracted at for exactly this reason.
If you convert a high prescription using an assumed 12 mm when the real figure was 15 mm, the error in the converted power at −10 D is around 0.25 D — a whole manufacturing step, produced entirely by an assumption. That is a good reason to treat the default in the box above as a placeholder rather than an answer.
Where This Sits Among the Other Optics Tools
This page does one conversion between two reference planes. To move between focal length and dioptric power, or to stack two thin lenses, use the diopter calculator. For imaging arithmetic — object distance, image distance and magnification — the thin lens equation calculator is the right tool, and the lens maker equation calculator derives a lens power from its surface curvatures and refractive index. Refraction at a boundary is handled by the Snell's law calculator and the index of refraction calculator.
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
- Treating the output as a prescription — it is a converted power and nothing else. Base curve, diameter, material, oxygen transmissibility, wearing schedule and product are all part of a contact lens prescription and none of them appear here.
- Applying the conversion to low powers — under about four dioptres the change is smaller than the step lenses are manufactured in, so converting changes nothing you could actually order.
- Assuming 12 mm — vertex distance depends on the frame and the face. At high powers a three-millimetre error in the assumption is worth a quarter dioptre in the answer.
- Converting the sphere and leaving the cylinder alone — both meridians carry power, so both convert. The cylinder at the corneal plane is the difference between the two converted meridians, not the original cylinder.
- Forgetting that plus and minus move opposite ways — minus powers weaken at the eye, plus powers strengthen. Getting the direction wrong doubles the error.
Related Free Tools From Arb Digital
Convert between focal length and dioptres with the diopter calculator, work imaging problems with the thin lens equation calculator, and derive power from surface geometry with the lens maker equation calculator. Handle boundaries and media with the Snell's law calculator and the index of refraction calculator, and size an optical instrument with the telescope magnification calculator. Every free tool Arb Digital publishes is listed on the free online tools hub.
Frequently Asked Questions
The power at the new plane equals the original power divided by one minus the distance times the original power, with the distance in metres and the power in dioptres. Written out: Fc = Fs / (1 - d x Fs). Reversing the direction simply reverses the sign of d.
Around four dioptres at a typical twelve-millimetre vertex distance. Below that the change is smaller than the quarter-dioptre step lenses are made in, so the conversion has no practical effect. Above it the change grows quickly, passing a full dioptre by about ten dioptres.
Because the sign of the product of distance and power flips the denominator either side of one. For a minus lens the denominator is greater than one, so the corneal-plane power is weaker. For a plus lens it is less than one, so the corneal-plane power is stronger.
No. This page converts a power between two reference planes. A contact lens prescription also specifies base curve, diameter, material, oxygen transmissibility, wearing schedule and the specific product it was fitted in, and those come from an examination and a fitting by an eye-care professional.
Yes, when the powers are high enough to matter. Each principal meridian converts separately, and the cylinder at the corneal plane is the difference between the two converted meridian powers. That is why the converted cylinder here is usually not the same number as the one you entered.
The measured one for the frame in question. Twelve millimetres is the conventional default used in textbooks and published tables, but the real distance depends on the frame and the wearer, and prescriptions for high powers often record the value the refraction was performed at.
Because the denominator reaches zero when the power equals the reciprocal of the vertex distance, about 83 dioptres at 12 millimetres. At that point the spectacle lens focuses light exactly at the corneal plane and no finite corneal-plane power reproduces it. It is a mathematical limit far outside any real prescription, but the calculator reports it rather than returning nonsense.
This tool is provided for educational use only. It performs one optical conversion and is not a prescription, a fitting, or medical advice of any kind. Only a qualified optometrist or ophthalmologist can prescribe or fit contact lenses, which are regulated medical devices; base curve, diameter, material, oxygen transmissibility, wearing schedule and product selection are clinical decisions made from an examination and cannot be derived from a converted power. Never wear, order or change lenses on the basis of a number produced by this page.
