Corneal Astigmatism from Keratometry Calculator
Corneal Astigmatism from Keratometry Calculator
Steep K minus flat K, in both cylinder forms with the axis each one takes, plus the radii behind the readings and why the 1.3375 that produced them is a convention and not a measurement.
K readings ⇄ corneal cylinder
KeratometryFlat K 43.00 D, steep K 44.50 D, flat meridian at 180° — with-the-rule astigmatism
The readings, the cylinder and the radius behind both
- which axis goes with which form
- a cylinder axis names the meridian in which the cylinder has NO power. A with-the-rule cornea is steeper vertically, so the eye is more myopic vertically, so the correcting minus cylinder must act vertically — which puts its axis horizontally, at the flat meridian. The plus-cylinder form of the same lens has its axis 90° away, at the steep meridian. The two forms are printed above and are exact transpositions
- 337.5, and what it assumes
- a keratometer measures a RADIUS and reports a POWER, and the bridge is a refractive index. The index essentially every instrument applies is 1.3375, so the power is (1.3375 − 1) ÷ r, which with r in millimetres is 337.5 ÷ r. A 7.5 mm radius is therefore 45.00 D exactly, and the conversion is reversible
- why 1.3375 is not the cornea’s refractive index
- the cornea’s own index is about 1.376 and the aqueous behind it about 1.336, so the posterior corneal surface carries NEGATIVE power. The 1.3375 keratometric index is a single number chosen to make an anterior-surface measurement stand in for the whole cornea’s power, and it rests, in the words of the study read for this page, on “the premise that the anterior and posterior corneal curvatures have a constant and linear relationship”. It is a convention, and a useful one, not a measurement
- where that premise breaks
- the same study measured posterior corneal astigmatism directly and found 0.31 ± 0.14 D against the 0.15 ± 0.10 D the keratometric assumption implies, in corneas whose keratometric astigmatism averaged 1.05 ± 0.68 D. The cause it identified is that the normal cornea is thicker vertically than horizontally (546.0 ± 31.8 against 542.6 ± 31.7 µm pericentrally), so the posterior surface is steeper vertically and, being of negative power, pushes the total astigmatism against-the-rule. Corneal refractive surgery reshapes the anterior surface and leaves the posterior one, which breaks the fixed relationship outright — and is the reason intraocular lens formulas that take a keratometric K fail after it
- Javal’s rule, and its stated scope
- refractive astigmatism estimated as 1.25 × the corneal astigmatism, plus 0.50 D of against-the-rule internal astigmatism. Both sources read give the same coefficient and constant, and the second states the rule for principal meridians at 0 and 90 degrees. So with-the-rule corneal astigmatism gives 1.25 C − 0.50 and against-the-rule gives 1.25 C + 0.50, and the estimate is NOT computed at an oblique axis: combining two cylinders at an oblique angle is a vector sum, and this calculator engine has no trigonometric function. The row above disappears rather than approximating it
- the internal astigmatism, measured
- the 0.50 D in Javal’s rule is a population average. The second source read measured it directly at −0.53 D in right eyes and −0.57 D in left, which it describes as close to Javal’s figure. A simplified form of the rule is attributed to Grosvenor and colleagues and is reported to fit better than the original; its coefficient is NOT printed here, because neither source read for this page states it
Worked example
Flat K 43.00 D, steep K 44.50 D, flat meridian at 180° — with-the-rule astigmatism
Corneal astigmatism = 44.50 − 43.00 = 1.50 D
The steep meridian is 90° from the flat one: 180 is above 90, so subtract — 90°. The cornea is steeper vertically, which is with-the-rule
Minus-cylinder form: −1.50 D at axis 180, the flat meridian. Plus-cylinder form: +1.50 D at axis 90, the steep meridian. The two are transpositions of each other and describe the same corneal surface
The radii. 337.5 ÷ 43.00 = 7.85 mm flat and 337.5 ÷ 44.50 = 7.58 mm steep. The difference is 0.27 mm, so 1.50 D of astigmatism is about a quarter of a millimetre of radius — which is why a keratometer has to be well calibrated
Javal's estimate. With-the-rule, so 1.25 × 1.50 − 0.50 = +1.38 D of expected refractive astigmatism. Against-the-rule — the same readings with the flat meridian at 90 — would give 1.25 × 1.50 + 0.50 = +2.38 D. The 1.00 D gap between the two is twice the internal cylinder, and it is the whole reason the rule needs to know the axis
The published worked case, reproduced. Readings of 44.00 and 45.00 at 090 give 1.00 D of astigmatism with the steep meridian at 090, which is the figure and the axis the source prints
Where the page refuses. Enter 45.00 as the flat and 43.00 as the steep and no answer appears: a negative astigmatism magnitude means the two fields are swapped. Set the flat meridian to 45° and the Javal row disappears while everything else stays — the rule is published for meridians at 0 and 90, and the oblique case is a vector sum this engine cannot form
Keratometric power and the radius behind it
| Radius (mm) | Keratometric power at index 1.3375 (D) | Power if the index were 1.3315 (D) | Difference |
|---|---|---|---|
| 6.80 | 49.63 | 48.75 | 0.88 D |
| 7.00 | 48.21 | 47.36 | 0.86 D |
| 7.20 | 46.88 | 46.04 | 0.83 D |
| 7.50 | 45.00 | 44.20 | 0.80 D |
| 7.80 | 43.27 | 42.50 | 0.77 D |
| 8.00 | 42.19 | 41.44 | 0.75 D |
| 8.40 | 40.18 | 39.46 | 0.71 D |
The two cylinder forms, and Javal’s estimate, by axis
| Flat K | Steep K | Flat meridian | Minus-cylinder form | Plus-cylinder form | Type | Javal estimate |
|---|---|---|---|---|---|---|
| 43.00 | 44.50 | 180 | −1.50 × 180 | +1.50 × 90 | With the rule | +1.38 D |
| 43.00 | 44.50 | 90 | −1.50 × 90 | +1.50 × 180 | Against the rule | +2.38 D |
| 43.00 | 44.50 | 45 | −1.50 × 45 | +1.50 × 135 | Oblique | Not computed |
| 41.00 | 43.00 | 10 | −2.00 × 10 | +2.00 × 100 | With the rule | +2.00 D |
| 42.00 | 46.00 | 120 | −4.00 × 120 | +4.00 × 30 | Against the rule | +5.50 D |
| 41.25 | 41.25 | 180 | 0.00 | 0.00 | Spherical cornea | −0.50 D |
A keratometer measures a radius and reports a power, and the bridge is a guess
Corneal astigmatism is the easiest calculation in this category and the one with the most concealed in it. The arithmetic is a subtraction: the steep keratometry reading minus the flat one is the corneal cylinder’s magnitude. What takes thought is the axis, because a cylinder axis names the meridian in which the cylinder has no power. A with-the-rule cornea is steeper vertically, so the eye is more myopic vertically, so the correcting minus cylinder must act vertically — and that puts its axis horizontal, at the FLAT meridian. The plus-cylinder form of the identical lens has its axis at the steep meridian instead. Both are printed above, and they are exact transpositions of each other, which is the check that the pair is right.
Underneath, a keratometer does not measure power at all. It measures the radius of curvature of the anterior corneal surface from the size of a reflected image, and converts it to dioptres with an assumed refractive index. The index essentially every instrument uses is 1.3375, which gives power = (1.3375 − 1)/r, or 337.5/r with the radius in millimetres — so a 7.5 mm cornea is reported as 45.00 D. The convention is not the cornea’s refractive index. The cornea’s own index is about 1.376 against about 1.336 for the aqueous behind it, which means the posterior corneal surface carries negative power; 1.3375 is a single number chosen so that measuring one surface stands in for the power of both, and it rests on the premise that the anterior and posterior curvatures keep a constant relationship.
They do not, quite. The study read for this page measured posterior corneal astigmatism directly and found roughly twice what the keratometric assumption implies (the figures are in the formula block above). Its explanation is anatomical: the normal cornea is a few microns thicker vertically than horizontally, so the posterior surface is steeper vertically, and because that surface has negative power it pulls the total astigmatism against-the-rule. The practical consequence is that keratometry systematically over-states with-the-rule astigmatism and under-states against-the-rule. Corneal refractive surgery breaks the premise altogether — it reshapes the front surface and leaves the back one — which is why intraocular lens formulas fed a keratometric K after LASIK produce the wrong answer, and why this category contains no lens power calculator.
Javal’s rule is the published bridge from corneal to refractive astigmatism: 1.25 times the corneal cylinder, plus 0.50 D of against-the-rule internal astigmatism from the lens. Both sources read give that coefficient and that constant, and one of them measured the internal cylinder directly at −0.53 D and −0.57 D, close to Javal’s figure. The rule is stated for principal meridians at 0 and 90 degrees, so this page computes it with-the-rule and against-the-rule and refuses at an oblique axis, where combining two cylinders is a vector addition needing a trigonometric term the engine does not have. A simplified form attributed to Grosvenor and colleagues is reported to fit better; its coefficient is not printed here, because neither source read states it and inventing it would be worse than its absence. For the single number a refraction is often summarised by, see the spherical equivalent; a cylinder’s vertex compensation has to be done meridian by meridian.
Frequently asked questions
How do I calculate corneal astigmatism from K readings?
Subtract the flat reading from the steep one. Readings of 43.00 and 44.50 give 1.50 D of corneal astigmatism. In minus-cylinder form that is −1.50 at the flat meridian’s axis; in plus-cylinder form, +1.50 at the steep meridian’s, 90° away.
Does the cylinder axis go with the flat or the steep meridian?
Both, depending on the form. The minus cylinder’s axis is the FLAT meridian and the plus cylinder’s is the STEEP one, because a cylinder axis names the meridian in which the cylinder has no power. A cornea steeper vertically needs minus power acting vertically, which is a minus cylinder with a horizontal axis.
How do I convert corneal radius in millimetres to dioptres?
Divide 337.5 by the radius in millimetres, and the inverse is the same division the other way. 7.5 mm is 45.00 D exactly. The 337.5 is (1.3375 − 1) × 1000, where 1.3375 is the keratometric index essentially every instrument applies.
Why is the keratometric index 1.3375 if the cornea’s index is 1.376?
Because 1.3375 is not meant to be the cornea’s index. A keratometer sees only the front surface, and the back surface — between a 1.376 cornea and a 1.336 aqueous — has negative power. The 1.3375 convention is one number chosen so that a front-surface measurement reports roughly the whole cornea’s power, and it assumes the two curvatures keep a fixed relationship. Measured directly, posterior astigmatism in normal corneas is 0.31 ± 0.14 D against the 0.15 ± 0.10 D the assumption implies.
What does Javal’s rule predict, and when does it not apply?
Refractive astigmatism of 1.25 times the corneal astigmatism plus 0.50 D of against-the-rule internal cylinder. On a 1.50 D with-the-rule cornea that is 1.38 D; against-the-rule, 2.38 D. It is published for principal meridians at 0 and 90 degrees, so this page does not apply it at an oblique axis: that is a vector addition of two cylinders and needs trigonometry the engine here does not have. A simplified version attributed to Grosvenor reportedly fits better, but its coefficient is in neither source read here and is therefore not printed.
Related calculators
References
- Open Exam Prep. Keratometer / Ophthalmometer: Calibration, Readings and Interpretation, Certified Lens Register study guide. open-exam-prep.com. Accessed 9 October 2026.
- Ueno Y, Hiraoka T, Miyazaki M, Ito M, Oshika T. Corneal thickness profile and posterior corneal astigmatism in normal corneas. Ophthalmology. 2015;122(6):1072-8.
- Javal’s rule. Wikipedia, citing Schwartz S. Geometric and Visual Optics. McGraw-Hill, 2002:219-21.
- Baghban Jaldian H, et al. Internal astigmatism measurement: testing a theory. J Ophthalmic Optometric Sci. 2024;8(2):1-6.
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/