Prism Dioptre to Degree Converter
Prism Dioptre to Degree Converter
1Δ is 0.573°, 30Δ is 16.70° and 100Δ is exactly 45°: the relation is an arctangent, not a ratio. Both directions, with the error in the 0.57° rule of thumb computed across the range.
Prism dioptres ⇄ degrees
Arctangent10Δ of prism, with 5° entered for the inverse
The conversion, and the rule of thumb it replaces
- why it is a tangent
- the prism dioptre is defined by a DISPLACEMENT, not by an angle. The two sources read state it as “one prism diopter (1 Δ) deflects a ray of light 1 cm at a distance of 1 meter” and “a prism of power 1Δ would produce 1 unit of displacement for an object held 100 units from the prism”. Displacement over distance is the tangent of the deviation, so Δ ÷ 100 is a tangent and the angle is its arctangent
- the exact relation, as published
- printed as “a = arc tan (P/100) or P = 100 tan a” in the one source read that gives it as readable text. The other two sources read give the DEFINITION but not the formula — one of them ends the sentence “this can be represented mathematically as:” with an image — and this page says so rather than implying three
- 0.57° per Δ, and where it comes from
- the rule of thumb is not arbitrary. A prism dioptre is almost exactly a CENTRAD, which is one hundredth of a radian and therefore 0.5729578°; the source read puts the gap at “only 0.003333133%” at small angles. The quoted 0.57 is that figure rounded down, and the rounding matters: the two versions of the rule disagree by 0.09° at 30Δ, and both are printed above
- the crossing at 12.50Δ
- because 0.57 is slightly below the centrad’s 0.5729578, the rounded rule starts out TOO SMALL while the arctangent’s curvature makes any linear rule grow too fast. The two effects cancel at 12.50Δ, where 0.57 × 12.5 = 7.125° and the exact value is 7.125° as well. Below that the rule under-states, above it the rule over-states, and a reader who knows only “the rule runs high” has it backwards for every small prism
- 100Δ is exactly 45°
- the one anchor in this relation that owes nothing to the small-angle region, and the one a reader can check without a calculator: a tangent of 1 is 45°. The published table read for this page gives the same value, and every other entry in that table was recomputed here from the tangent and agrees
Worked example
10Δ of prism, with 5° entered for the inverse
10Δ displaces the image 10 cm at 1 m, so the tangent of the deviation is 10 ÷ 100 = 0.10
arctan(0.10) = 5.711°
The rule of thumb. 0.57 × 10 = 5.700°, which is 0.011° LOW — not high. Treating the prism dioptre as a centrad instead gives 5.730°, which is 0.019° high. The two approximations straddle the right answer at this size
The inverse. 100 × tan(5°) = 8.75Δ. Note that it is not 5 ÷ 0.57 = 8.77Δ, and the difference is already visible at the second decimal place
The displacement. 10Δ shifts the image 10 cm at one metre and 60 cm at the six metres of an acuity chart, which is the definition restated and is the reason the unit exists
Where the rule fails. At 30Δ the exact angle is 16.699° and the rule gives 17.100°, an error of 0.401°; at 40Δ the error is a full degree; at 100Δ the rule gives 57° where the answer is exactly 45°
And it is not linear. Doubling 10Δ to 20Δ does not double the angle: 5.711° becomes 11.310°, a factor of 1.98. Doubling 50Δ to 100Δ gives a factor of 1.69
Prism dioptres to degrees, with both linear rules and their errors
| Δ | Exact (°) | 0.57° rule (°) | Error (°) | Centrad rule (°) | Error (°) |
|---|---|---|---|---|---|
| 1 | 0.573 | 0.570 | −0.003 | 0.573 | +0.000 |
| 5 | 2.862 | 2.850 | −0.012 | 2.865 | +0.002 |
| 10 | 5.711 | 5.700 | −0.011 | 5.730 | +0.019 |
| 12.50 | 7.125 | 7.125 | 0.000 | 7.162 | +0.037 |
| 15 | 8.531 | 8.550 | +0.019 | 8.594 | +0.064 |
| 20 | 11.310 | 11.400 | +0.090 | 11.459 | +0.149 |
| 25 | 14.036 | 14.250 | +0.214 | 14.324 | +0.288 |
| 30 | 16.699 | 17.100 | +0.401 | 17.189 | +0.489 |
| 40 | 21.801 | 22.800 | +0.999 | 22.918 | +1.117 |
| 50 | 26.565 | 28.500 | +1.935 | 28.648 | +2.083 |
| 100 | 45.000 | 57.000 | +12.000 | 57.296 | +12.296 |
Degrees to prism dioptres, and what the scale does at the ends
| Degrees | Δ, exact | Δ on the 0.57 rule | Comment |
|---|---|---|---|
| 1 | 1.746 | 1.754 | The two agree to three parts in a thousand |
| 2 | 3.492 | 3.509 | Still interchangeable at this size |
| 5 | 8.749 | 8.772 | 0.02Δ apart, below measurement resolution |
| 10 | 17.633 | 17.544 | Now 0.09Δ apart, and the sign has reversed |
| 15 | 26.795 | 26.316 | 0.48Δ apart |
| 20 | 36.397 | 35.088 | 1.31Δ apart |
| 30 | 57.735 | 52.632 | 5.10Δ apart — the rule is unusable |
| 45 | 100.000 | 78.947 | Exactly 100Δ, and the rule is 21Δ low |
| 90 | infinite | 157.895 | The scale has no value here at all |
| 135 | −100.000 | 236.842 | Published as NEGATIVE prism dioptres |
| 180 | 0.000 | 315.789 | Back to zero, which is the scale’s other absurdity |
A prism dioptre is a tangent, which is why the rule of thumb has a crossing point
The prism dioptre is defined by displacement and not by angle: one prism dioptre moves the image one centimetre at one metre. That makes Δ ÷ 100 the tangent of the deviation, so the angle is arctan(Δ ÷ 100) and the inverse is 100 × tan(degrees). The relation is exact, published, and not linear. One hundred prism dioptres is exactly forty-five degrees, which is the anchor worth remembering because it needs no calculator and it kills the linear rule outright — a rule that gives 0.57° per dioptre predicts 57° there.
The rule of thumb is not folklore, though. A prism dioptre is very nearly a centrad, which is one hundredth of a radian and so 0.5729578°; the source read for this page measures the gap at three thousandths of a per cent at small angles. The familiar 0.57 is that number rounded down, and the rounding produces the most surprising fact on this page. Because 0.57 starts out too small while any linear rule eventually runs too fast, the two errors cancel at 12.50 prism dioptres — below which the rounded rule UNDER-states the angle, and above which it over-states it. A reader who has learned that the rule “runs high” has the sign backwards for every small prism they will ever measure.
Where it stops being usable depends on what you will tolerate, and all three figures are this page’s own arithmetic. The 0.57° rule stays within a tenth of a degree to 20.5Δ, within a quarter of a degree to 26.1Δ, within half a degree to 32.1Δ, and reaches a full degree at 40.0Δ. The centrad version is slightly worse because it has no crossing to help it: it is already 0.49° out at 30Δ against the rounded rule’s 0.40°. In relative terms the error passes 1% between 20Δ and 25Δ, and reaches 26.7% at 100Δ.
Two things this page deliberately does not do. It does not interpret a deviation: the number of prism dioptres a strabismus measures, and what follows from it, is a clinical matter and no arithmetic settles it. And it does not pretend the unit is well-behaved at the top of its range — the source read says plainly that ninety degrees is infinite prism dioptres and that angles beyond ninety come out negative, which is a property of the unit rather than of this conversion. For the prism a lens induces rather than the prism a patient has, see Prentice’s rule, which uses this same definition from the other end: the induced prism in dioptres is the image displacement in centimetres per metre. The power that rule needs in an oblique meridian is on the oblique meridian page.
Frequently asked questions
How many degrees is one prism dioptre?
0.5729°, and the exact relation is arctan(Δ ÷ 100) rather than a multiplication. The familiar rule of 0.57° per prism dioptre is that figure rounded, and at 1Δ it is 0.003° LOW. It only becomes a high estimate above 12.50Δ.
How do I convert prism dioptres to degrees?
Take the arctangent of the prism dioptres divided by 100, because a prism dioptre is a displacement of one centimetre at one metre and therefore a tangent of one in a hundred. 10Δ is 5.711°, 30Δ is 16.699° and 100Δ is exactly 45°. The inverse is 100 × tan(degrees).
Is the 0.57 degrees per prism dioptre rule accurate?
Up to about 20 prism dioptres, yes — it is within a tenth of a degree to 20.5Δ. It reaches a quarter of a degree of error at 26.1Δ, half a degree at 32.1Δ and a full degree at 40.0Δ. At 30Δ it gives 17.10° against an exact 16.699°. Above about 30 prism dioptres it should not be used.
Why is 100 prism dioptres 45 degrees and not 57?
Because 100Δ means the image moves 100 cm at 100 cm, and an equal rise and run is a tangent of 1, which is 45° exactly. The 57° a linear rule predicts is the error the rule accumulates over that range — twelve degrees, or 26.7%.
Does doubling the prism double the angle?
No, and that is the test that distinguishes this conversion from a unit factor. Going from 10Δ to 20Δ multiplies the angle by 1.98, not 2; going from 50Δ to 100Δ multiplies it by 1.69. The shortfall grows, and at 90° the prism dioptre scale becomes infinite.
Related calculators
References
- Bicas HEA. A new unity for angular measurements in strabismus [published under the heading “Advantages and disadvantages of the prism-dioptre unity”]. Arq Bras Oftalmol. 2014;77(5):275-9. (Prints a = arc tan (P/100) and P = 100 tan a as text, with two conversion tables whose values were independently recomputed here.)
- Open Exam Prep. Section 2.3: Prism and Prentice’s Rule, ABO Advanced study guide. open-exam-prep.com. Accessed 10 October 2026.
- Prism correction. Wikipedia, citing Carlton J. Frames and Lenses. 2000:53ff. (The definition is text; the article’s formula is not, and was not used.)
- Open Exam Prep. Section 15.1: ANSI Z80.1 Prescription Tolerances, National Opticianry Competency Examination study guide. open-exam-prep.com. Accessed 10 October 2026. (ANSI Z80.1 itself is a purchased standard and was not read.)
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/