Prentice's Rule Calculator: Induced Prism
This calculator applies Prentice's rule to find the prism a lens induces when the eye looks through a point away from its optical center. Enter the sphere, cylinder, axis, and how far and which way the optical center is displaced from the pupil. It returns the prism in prism diopters, its base direction, and a quick check against the ANSI Z80.1-2020 limits.
Induced Prism (Prentice’s Rule)
Calculate the prism induced when looking through a sphero-cylindrical lens off its optical center. Uses the effective power in the meridian of decentration and screens the result against the ANSI Z80.1 unwanted-prism limits.
Above ±2.75 D, ANSI Z80.1-2020 checks placement instead of prism: the distance between the prism reference points must be within ±2.5 mm of the ordered PD.
Formula & tolerances
Power in meridian θ: Pθ = S + C · sin²(θ − axis)
Horizontal decentration → use P at θ=0° = S + C · sin²(axis)
Vertical decentration → use P at θ=90° = S + C · cos²(axis)
Δ = (decentration in cm) × |effective power|
Screens one lens against the ANSI Z80.1-2020 mounted-pair limits (0.67Δ horizontal up to ±2.75 D, 0.33Δ vertical up to ±3.375 D; above those powers the standard checks placement in mm).
Prentice's rule formula
The formula is Δ = c × F, where Δ is the induced prism in prism diopters, c is the decentration in centimeters, and F is the lens power in diopters. Convert millimeters first: 5 mm is 0.5 cm.
For a sphero-cylinder, F is the power in the meridian along which the decentration runs, not the sphere. The calculator uses P = S + C × sin²(θ − axis): horizontal decentration takes S + C × sin²(axis), vertical decentration takes S + C × cos²(axis).
Base direction: on a plus lens the base points the same way the optical center is displaced; on a minus lens it points the opposite way. The base always points toward the thicker part of the lens.
Worked examples
Plus sphere: +2.50 D, optical center 5 mm nasal of the pupil (In). Δ = 0.5 × 2.50 = 1.25Δ, base in. The calculator shows 1.25Δ BI and flags it as over its 0.67Δ screen. Up to ±2.75 D the standard applies that limit to the pair: a pair made this way carries 2.50Δ base in.
Sphero-cylinder: -2.00 -1.50 × 180 with the optical center 3 mm below the pupil (Down). The vertical meridian power is -2.00 + (-1.50) = -3.50 D, so Δ = 0.3 × 3.50 = 1.05Δ. It is a minus lens, so the base is opposite the displacement: 1.05Δ BU.
Solving for decentration: to create 1.00Δ base in with a -4.00 D lens, c = Δ ÷ F = 1 ÷ 4 = 0.25 cm, so move the optical center 2.5 mm out. Entering -4.00, 2.5 mm, Out returns 1.00Δ BI.
Oblique axis: -1.00 -2.00 × 45, optical center 4 mm out. Horizontal power = -1.00 + (-2.00 × sin²45°) = -2.00 D, so 0.80Δ BI. The oblique cylinder also adds 0.40Δ of vertical prism, which the calculator does not report.
When to use it, and the ANSI Z80.1-2020 check
Use it to see how much prism a PD or optical-center error creates, to find the decentration that produces a prescribed prism, and to estimate vertical imbalance at reading level. Its badge screens one lens against 0.67Δ horizontal and 0.33Δ vertical, which ANSI Z80.1-2020 applies to the mounted pair: net horizontal prism and the vertical difference between the eyes.
Both prism limits stop at a cutoff power in the meridian checked. Horizontally, 0.67Δ applies up to ±2.75 D; above that, the distance between the prism reference points (PRPs) must be within ±2.5 mm of the ordered PD. Vertically, 0.33Δ applies up to ±3.375 D; above that, the PRP heights must be within 1.0 mm of each other. See the full ANSI tolerances.
Common mistakes
Putting millimeters into the formula (5 × 2.50 = 12.50Δ instead of 1.25Δ). Using the sphere when the decentration runs along the cylinder's power meridian. Reversing the base rule for minus lenses. And mixing up the eye and the optical center: if the eye reads 10 mm below the optical center, the optical center is 10 mm up from the line of sight, so choose Up.
Prentice's rule on the ABO exam
These are also the common forms of Prentice's rule problems in ABO prep. For example, a wearer with OD -2.00 D and OS -4.50 D who reads 8 mm below the optical centers gets 1.60Δ BD and 3.60Δ BD, a 2.00Δ imbalance. A slab-off to correct it goes on the more minus lens, here the left. Work problems by hand, check them here, then review the Prism study guide and take the free ABO practice test.
Frequently asked questions
What is Prentice's rule?
Prentice's rule says the prism induced at a point on a lens equals its distance from the optical center in centimeters times the lens power in diopters: Δ = c × F. It explains why a wrong PD or a misplaced optical center gives the wearer prism they were not prescribed.
Is the base direction the same for plus and minus lenses?
No. On a plus lens the base points the same way the optical center is displaced from the pupil; on a minus lens it points the opposite way. With the optical center 2 mm out, a -5.00 D lens gives 1.00Δ base in and a +5.00 D lens gives 1.00Δ base out.
Which power do I use for a cylinder lens?
Use the power in the meridian along which the decentration runs. For -2.00 -1.50 × 180, horizontal decentration uses -2.00 D and vertical decentration uses -3.50 D. For an oblique axis, the power in a meridian θ is S + C × sin²(θ − axis).
How much unwanted prism does ANSI Z80.1-2020 allow?
For a mounted pair with no prism prescribed, ANSI Z80.1-2020 allows 0.67Δ of net horizontal prism when the horizontal power is 0.00 to ±2.75 D, and 0.33Δ of vertical prism difference between the lenses when the vertical power is 0.00 to ±3.375 D. Above those powers it checks placement in millimeters instead: the distance between the prism reference points must be within ±2.5 mm of the ordered PD horizontally, and their heights within 1.0 mm of each other vertically. A 2025 edition has been announced with changes to the prism tolerances, so confirm which edition your exam follows.
Related tools and study guides
Ready to try the math on exam-style questions?
Take a free ABO practice test: instant scoring and an explanation after every answer. No card required.
Start the free practice test →