Prism diopter calculator
Prism, degrees and shift at a distance, plus the prism a lens makes when you look off its center.
For general information only — not medical advice. This page does educational arithmetic with optics formulas; it is not a prescription. Prism for a patient must be prescribed by an eye-care professional.
Prism and degrees: enter the value; the distance is only used to show the sideways shift. For a shift, enter how far the target moved and the distance it was at, in the same unit.
Use the power in the meridian you look through. Fill in the optional prism to get the center shift that would produce it.
What a prism diopter measures
A prism bends light toward its base. The strength of that bend is given in prism diopters, written Δ: a prism of 1Δ moves the image of an object 1 centimeter sideways when the object is 1 meter away. Opticians and eye-care professionals use it for prism ground into glasses, for the misalignment of the eyes, and for the unwanted prism a lens creates when the wearer looks through a point away from its optical center.
Formulas, variable by variable
- Prism and angle: Δ = 100 × tan(θ), so θ = arctan(Δ ÷ 100). Here θ is the angle through which the light is bent.
- Shift at a distance: shift = Δ × distance ÷ 100, in the same unit as the distance.
- Prentice’s rule: Δ = c × F, with c the distance from the lens’s optical center in centimeters (millimeters ÷ 10) and F the lens power in diopters. The reverse, c = 10 × Δ ÷ F in millimeters, gives the decentration that yields a wanted prism.
Worked examples
One prism diopter is 0.573°, and five degrees is 100 × tan(5°) = 8.75Δ. A 8.75Δ prism moves a target by 1.40 inches at 16 inches. For Prentice’s rule, take a +3.00 D lens and look 5 mm from its center: 0.5 cm × 3.00 D = 1.5Δ, with the base toward the optical center. To get 1.5Δ from this lens the center must sit 5.0 mm from the line of sight. The 20/20 magazine course also works two pairs of lenses. For Carlos, whose right lens is −2.00 D and left −3.00 D, with centers 4 mm and 5 mm off, the lenses give 0.8Δ and 1.5Δ, both base out, and they add to 2.3Δ base out. For Derek, +4.00 D at 2 mm and +5.00 D at 1 mm give 0.8Δ base out and 0.5Δ base in, which partly cancel to 0.3Δ.
Prism, angle and shift side by side
| Prism (Δ) | Angle (°) | Shift at 1 m (cm) | Shift at 40 in (in) |
|---|---|---|---|
| 1 | 0.57 | 1.0 | 0.4 |
| 2 | 1.15 | 2.0 | 0.8 |
| 5 | 2.86 | 5.0 | 2.0 |
| 10 | 5.71 | 10.0 | 4.0 |
| 20 | 11.31 | 20.0 | 8.0 |
| 50 | 26.57 | 50.0 | 20.0 |
| 100 | 45.00 | 100.0 | 40.0 |
All rows come from Δ = 100 × tan θ and shift = Δ × distance ÷ 100. Up to about 10Δ the angle is close to 0.57 degrees per prism diopter; at 100Δ the angle is exactly 45 degrees, where that shortcut would give 57.
Limits and common mistakes
- Degrees are not proportional. The tangent bends the scale: 100Δ is 45 degrees. For small prescriptions the 0.57 factor is close, but it drifts at large angles.
- Prentice’s rule is a thin-lens rule. It uses the power in the direction of the shift. In a lens with astigmatism, that is the power of the meridian you look along, not the sphere.
- Direction. Plus lenses put the base toward the optical center and minus lenses away from it; mixing these up reverses base in and base out. The magazine’s examples (+3.00 D with 1.5Δ base in, −3.00 D with 1.5Δ base in) show both lens signs reaching base in by moving the centers opposite ways.
- Two eyes. The prism of the right and left lenses add or cancel depending on direction, as in the Carlos and Derek examples above. The calculator handles one lens at a time.
Common questions
What is a prism diopter?
One prism diopter (Δ) bends a ray of light so that it lands 1 centimeter off to the side at a distance of 1 meter. Two prism diopters shift it 2 cm at 1 m, and so on. As an angle, 1Δ is 0.573 degrees, because Δ = 100 × tan(angle).
How do I convert prism diopters to degrees?
Divide by 100, then take the inverse tangent: angle = arctan(Δ ÷ 100). For 10Δ that is 5.71 degrees. Going the other way, 5 degrees is 100 × tan(5°) = 8.75Δ. For small angles degrees are about 0.57 times the prism diopters, but the relation is not linear: 100Δ is exactly 45 degrees, not 57.
What is Prentice’s rule?
The prism a lens produces at a point away from its optical center equals the distance from that center in centimeters times the lens power in diopters: Δ = c × F. A +3.00 D lens looked through 5 mm (0.5 cm) from its center gives 1.5Δ. The 20/20 magazine course on prism uses the same numbers.
Which way does the prism point?
In a plus lens the base of the prism points toward the optical center; in a minus lens it points away from it. This is why moving the optical centers of a plus lens inward produces base-in prism for the wearer, while moving those of a minus lens outward does the same. The calculator states the direction relative to the optical center and leaves the wearer-specific words (base in, base out, up, down) to the optician.
How much does a shift of 1Δ look like in real life?
A 1Δ displacement is 1 cm at 1 meter, 0.4 inches at 40 inches, and about 2.4 inches at 20 feet, since 20 feet is 240 inches. The sideways shift doubles when the distance doubles, while the angle stays the same.
Can this tell me how much prism I need?
No. It only does the arithmetic between units and the thin-lens rule. How much prism, if any, a person needs is a clinical measurement and prescription decision.
Related pages: the Snell’s law calculator for the refraction behind a prism, the thin lens calculator, the lens thickness calculator and the vertex distance calculator.
For general information only — not medical advice. This page does educational arithmetic; prism for a patient must be prescribed by an eye-care professional. Sources: K. Rands, “The Many Facets of Prism in Ophthalmic Lenses, Part 2,” 20/20 magazine (Jobson Medical Information), for Prentice’s rule and the Carlos, Derek and +3.00 D examples; the definition of the prism diopter and Δ = 100 × tan θ as given in the “Prism correction” article of Wikipedia (secondary source), with degrees computed by us. How we check our calculators.