Vertex Distance Compensation Calculator
Vertex Distance Compensation Calculator
A +10.00 D spectacle lens at 12 mm needs +11.36 D at the eye; a −10.00 D needs only −8.93. F ÷ (1 − dF), in both directions, with the point at which the change reaches a quarter dioptre computed rather than quoted.
Spectacle ⇄ contact lens power
Effective power+10.00 D measured at a 12 mm vertex distance, fitted at 0 mm — a spectacle lens becoming a contact lens
Formula, and the sign of d
- d, and the inversion this page exists to prevent
- d is the CHANGE in vertex distance, not a vertex distance, and it is POSITIVE when the lens moves closer to the eye. Written the other way round the formula still returns a plausible number: at 12 mm a +10.00 D lens gives +11.36 D the right way and +8.93 D the wrong way. Both are valid lens powers and neither looks like an error, which is why the wrong-direction answer is printed above beside the right one
- metres, not millimetres
- d must be in metres: 12 mm is 0.012. The millimetre form of the same formula is F ÷ (1 − d(mm) × F ÷ 1000), which is what the second source read for this page prints. Using millimetres without the division by 1000 makes the denominator negative and the sign of the answer flips
- back vertex distance
- the distance from the BACK surface of the lens to the front of the cornea, which is also why spectacle powers are specified as back vertex powers. Conventional refracting distance is 12 to 14 mm; fitted distances run about 10 to 17 mm
- the ±4.00 D convention, and what the arithmetic says
- both sources give ±4.00 D as the power from which compensation is applied, and one of them is internally inconsistent about whether 4.00 D itself qualifies — its summary says “over ±4.00 D” and its body says “≥ ±4.00 D”. Computed from the formula at a 12 mm vertex, the change first reaches 0.25 D at +4.44 D and at −4.69 D, so the convention is slightly early and early by different amounts in the two signs. The ±4 D figure is the published rule of thumb; these two are this page’s own arithmetic
- where the formula stops describing a lens
- when 1 − dF reaches zero the compensated power is infinite and beyond it the sign flips, at F = 1/d — about +83 D at a 12 mm move. No spectacle lens reaches it, but the ±40 D and 30 mm ranges this page accepts do, and it refuses there rather than printing the sign-flipped number
Worked example
+10.00 D measured at a 12 mm vertex distance, fitted at 0 mm — a spectacle lens becoming a contact lens
d = (12 − 0) ÷ 1000 = +0.012 m, positive because the lens is moving toward the eye
1 − dF = 1 − (0.012 × 10.00) = 1 − 0.12 = 0.88
Fnew = 10.00 ÷ 0.88 = +11.36 D, so the contact lens needs 1.36 D MORE plus than the spectacle lens
Now the other sign, same geometry. −10.00 D at 12 mm: 1 − (0.012 × −10.00) = 1.12, and −10.00 ÷ 1.12 = −8.93 D — 1.07 D LESS minus. The second source read for this page prints exactly −8.93 D for this case. Note that the two changes are not equal: 1.36 against 1.07, because the denominator moves the opposite way in the two signs
And back again, to prove the formula inverts. Take the −8.93 D at 0 mm and move it out to 12 mm: d = (0 − 12) ÷ 1000 = −0.012, 1 − (−0.012 × −8.93) = 0.893, and −8.93 ÷ 0.893 = −10.00 D. The round trip is exact to machine precision and is asserted for forty power-and-distance pairs
The published worked cases, reproduced. +11.50 D refracted at 15 mm and fitted at 11 mm: d = +0.004, 11.50 ÷ 0.954 = +12.05 D, which the source rounds to +12.00. And −9.00 D refracted at 14 mm fitted at 10 mm: d = +0.004, −9.00 ÷ 1.036 = −8.69 D, rounded to −8.75
The identity. Set both vertex distances equal and the answer is the power entered, for every power. That is also the one case where a formula with d's sign reversed is right, which is why it is a useless test on its own
Compensated power at a 12 mm move to the corneal plane
| Spectacle power at 12 mm | At the eye | Change | Nearest 0.25 D step | Reaches 0.25 D? |
|---|---|---|---|---|
| +16.00 | +19.80 | +3.80 | +19.75 | Yes |
| +10.00 | +11.36 | +1.36 | +11.25 | Yes |
| +4.44 | +4.69 | +0.25 | +4.75 | Exactly |
| +4.00 | +4.20 | +0.20 | +4.25 | No |
| +2.00 | +2.05 | +0.05 | +2.00 | No |
| −2.00 | −1.95 | +0.05 | −2.00 | No |
| −4.00 | −3.82 | +0.18 | −3.75 | No |
| −4.69 | −4.44 | +0.25 | −4.50 | Exactly |
| −10.00 | −8.93 | +1.07 | −9.00 | Yes |
| −20.00 | −16.13 | +3.87 | −16.25 | Yes |
The same power at four vertex distances
| Vertex distance | +10.00 D becomes | −10.00 D becomes | +5.00 D becomes | −5.00 D becomes |
|---|---|---|---|---|
| 0 mm (no move) | +10.00 | −10.00 | +5.00 | −5.00 |
| 8 mm | +10.87 | −9.26 | +5.21 | −4.81 |
| 10 mm | +11.11 | −9.09 | +5.26 | −4.76 |
| 12 mm | +11.36 | −8.93 | +5.32 | −4.72 |
| 14 mm | +11.63 | −8.77 | +5.38 | −4.67 |
| 17 mm | +12.05 | −8.55 | +5.46 | −4.61 |
Why a lens changes power when you move it, and which way
A lens does not have one power at the eye; it has the power its focal point implies from wherever it is sitting. Move it closer to the cornea and its second focal point moves with it, so the vergence arriving at the eye changes. The compensation is exact and closed form: the new power is the old power divided by one minus the product of the distance moved, in metres, and the old power. Everything difficult about it is the sign of that distance.
The direction flips with the sign of the lens, and it is worth stating as two separate facts rather than one. Moving a lens toward the eye shifts its required power in the plus direction, whatever its sign. For a plus lens that means MORE plus: a +10.00 D spectacle lens at 12 mm needs +11.36 D at the corneal plane. For a minus lens it means LESS minus: a −10.00 D spectacle lens needs only −8.93 D. The two changes are not the same size — 1.36 D against 1.07 D — because the denominator 1 − dF shrinks below one for a plus power and grows above one for a minus power. A rule of thumb calibrated on myopes under-states the correction for hyperopes.
The convention is to compensate from ±4.00 D, which both sources read for this page give and one of them states twice with different inclusivity — “over ±4.00 D” in its summary and “≥ ±4.00 D” in its body. The argument behind the figure is a rounding argument: below it the change is smaller than the 0.25 D step lenses are made in, so compensating would not alter what is ordered. Computed from the formula at a 12 mm vertex, the change first reaches a quarter dioptre at +4.44 D and at −4.69 D, so the published threshold is slightly early and early by different amounts in the two signs. That is not a correction to the convention — rounding to a quarter dioptre can cross at ±4 D depending on where the power already sits relative to a step — but it is the arithmetic, and it is printed here rather than asserted.
Three practical points the formula does not say on its own. First, a sphero-cylindrical prescription must be compensated meridian by meridian at high powers: compensate the sphere and the sphere-plus-cylinder separately and re-form the prescription, because the steeper meridian changes more than the flatter one and compensating the sphere alone leaves the cylinder wrong. Second, d is a back vertex distance, measured from the back surface of the lens, which is also why spectacle lenses are specified by back vertex power. Third, testing the arithmetic at d = 0 proves nothing: that is the one case where the formula with the sign of d reversed, or with d left in millimetres, returns the right answer. The wrong-direction result is printed above beside the right one, because at +10.00 D and 12 mm the two are +11.36 and +8.93 — both perfectly ordinary lens powers, and nothing about either looks like a mistake. For the spherical equivalent of a compensated prescription, compensate first and summarise afterwards; the two operations do not commute.
Frequently asked questions
What is the vertex distance compensation formula?
New power = old power ÷ (1 − d × old power), with d the change in vertex distance in metres, positive when the lens moves closer to the eye. A +10.00 D lens refracted at 12 mm becomes 10 ÷ (1 − 0.012 × 10) = +11.36 D at the cornea.
Does a contact lens need more or less power than the glasses?
It depends on the sign. A plus lens needs MORE plus at the eye: +10.00 D of spectacle becomes +11.36 D of contact lens at a 12 mm vertex. A minus lens needs LESS minus: −10.00 D of spectacle becomes −8.93 D. Moving a lens toward the eye always shifts the required power in the plus direction, which reads as stronger for a plus lens and weaker for a minus one.
At what power does vertex distance start to matter?
The published convention is ±4.00 D, which both sources read for this page give — though one states it as “over ±4.00” in its summary and “≥ ±4.00” in its body. The reasoning is that below that the change is under the 0.25 D step lenses are made in. Computed from the formula at a 12 mm vertex the change first reaches 0.25 D at +4.44 D and at −4.69 D.
Why is d negative when the lens moves away from the eye?
Because d is defined as the old vertex distance minus the new one. Moving away makes the new distance larger, so d is negative, the denominator 1 − dF grows for a plus lens and the compensated power falls. Getting the sign backwards returns a plausible lens power rather than an error: at +10.00 D and 12 mm the right answer is +11.36 and the wrong one is +8.93.
Do I compensate the sphere and the cylinder separately?
At high powers, yes. Each principal meridian has its own power and compensates by its own amount, so the correct method is to compensate the sphere and the sphere-plus-cylinder as two separate powers and then re-form the prescription from the compensated pair. Compensating the sphere and carrying the original cylinder across is an approximation whose error falls on the cylinder.
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References
- Open Exam Prep. Section 3.3: Vertex Distance Compensation, National Opticianry Competency Examination study guide. open-exam-prep.com. Accessed 9 October 2026.
- Open Exam Prep. Vertex Distance, Sagittal Depth and Spherical Equivalent, Certified Lens Register study guide. open-exam-prep.com. Accessed 9 October 2026.
- Laramy-K Optical. Transposing Prescriptions. laramyk.com dispensing education. Accessed 9 October 2026.
Not medical advice. For healthcare professionals and education. Reference intervals vary by laboratory and assay — always use your own laboratory's. Never base a dose or a treatment decision on this page alone. Full disclaimer at calcengines.com/disclaimer/