Spherical Equivalent Calculator
Spherical Equivalent Calculator
Sphere plus half the cylinder. One number for a refraction with two principal meridians — and the number epidemiology counts myopia with, where moving the threshold 0.50 D nearly doubles the prevalence.
Sphere + cyl ÷ 2
Refraction summarySphere −2.00 D, cylinder −1.00 D
Formula
- why half
- a sphero-cylindrical lens has two principal meridians, one with the power of the sphere and one with the sphere plus the cylinder. The spherical equivalent is their arithmetic mean, and the mean of x and x + c is x + c/2. It is the spherical power that puts the circle of least confusion on the retina, which is why it is the single sphere that best replaces the pair
- the sign is load-bearing
- a −1.00 cylinder shifts the equivalent 0.50 D more minus and a +1.00 cylinder shifts it 0.50 D more plus. Entering a cylinder’s magnitude without its sign produces an error of the whole cylinder, which on a −2.00 cylinder is 2.00 D
- invariant under transposition
- transposing adds the cylinder to the sphere and flips the cylinder’s sign, so the new equivalent is (s + c) + (−c)/2 = s + c/2 — the same number. Both forms are printed above, and they must agree exactly; if they do not, the prescription was mis-entered
- what it is not
- it is not the prescription, and it is not a lens anybody should wear in place of the cylinder. It is a summary used where one number per eye is needed: epidemiology, comparing refractions across a cohort, setting a trial lens before the cylinder is refined
- the myopia thresholds
- Cumberland, Bountziouka and Rahi applied three definitions to the same autorefraction data in the 1958 British birth cohort (1,985 people, measured at age 44 to 45): myopia at −1.00 D, −0.75 D and −0.50 D. Mild-myopia prevalence came out at 28%, 36% and 47%, and emmetropia at 50%, 42% and 31%. Their conclusion is quoted: “Even small changes in the threshold definition of myopia (±0.25D) can significantly affect the conclusions” of studies
Worked example
Sphere −2.00 D, cylinder −1.00 D
Half the cylinder = −1.00 ÷ 2 = −0.50 D
Spherical equivalent = −2.00 + (−0.50) = −2.50 D
The two principal meridians are −2.00 D and −2.00 + (−1.00) = −3.00 D, and −2.50 is their midpoint
Transposed, the same answer. −2.00 −1.00 × 180 transposes to −3.00 +1.00 × 90. Its equivalent is −3.00 + (+1.00 ÷ 2) = −2.50 D. The agreement is exact and is the cheapest available check on a transposition
Where the wrong versions diverge. Half the sphere plus the cylinder would give −1.00 + (−1.00) = −2.00; sphere minus half the cylinder would give −1.50; sphere plus cylinder would give −3.00. All four agree only when the cylinder is zero, which is why a test at plano proves nothing
And the threshold question, on these numbers. −2.50 D is myopic under all three published definitions and sits in the mild stratum of each. Change the sphere to −0.50 and the cylinder to −0.50: the equivalent is −0.75 D, which is myopic under the −0.75 and −0.50 models and emmetropic under the −1.00 model. One refraction, two different published answers
A plus example. Sphere +1.00, cylinder +3.00 gives +1.00 + 1.50 = +2.50 D, with meridians of +1.00 and +4.00. Sphere 0.00 with cylinder −4.00 gives −2.00 D, with meridians of 0.00 and −4.00 — a refraction that is neither myopic nor hypermetropic in one meridian and 4.00 D myopic in the other, summarised as a 2.00 D myope
One cohort, three published myopia definitions
| Definition of mild myopia | Mild myopia | Emmetropia | Moderate to high myopia (≤ −3.00 D) | Hypermetropia (≥ +1.00 D) |
|---|---|---|---|---|
| Model 1: SE −1.00 to −2.99 D | 28% (95% CI 26–30) | 50% | 13% | 9.0% |
| Model 2: SE −0.75 to −2.99 D | 36% (95% CI 34–39) | 42% | 13% | 9.0% |
| Model 3: SE −0.50 to −2.99 D | 47% (95% CI 45–49) | 31% | 13% | 9.0% |
The same refraction under each cylinder form
| Minus-cylinder form | Plus-cylinder form | Meridian powers | Spherical equivalent |
|---|---|---|---|
| −2.00 −1.00 × 180 | −3.00 +1.00 × 90 | −2.00 and −3.00 | −2.50 D |
| −5.00 −2.50 × 10 | −7.50 +2.50 × 100 | −5.00 and −7.50 | −6.25 D |
| +1.00 −0.75 × 45 | +0.25 +0.75 × 135 | +1.00 and +0.25 | +0.625 D |
| 0.00 −4.00 × 90 | −4.00 +4.00 × 180 | 0.00 and −4.00 | −2.00 D |
| −1.50 0.00 | −1.50 0.00 | −1.50 and −1.50 | −1.50 D |
One number for a two-meridian refraction, and what it throws away
A sphero-cylindrical lens has two principal meridians at right angles, one carrying the power of the sphere and the other the sphere plus the cylinder. The spherical equivalent is their arithmetic mean, which is why it is the sphere plus half the cylinder: the mean of x and x + c is x + c/2. Optically it is the spherical power that places the circle of least confusion — the point between the two focal lines where the blur is smallest and symmetrical — on the retina. That is what makes it the single sphere that best stands in for the pair, and it is also why it corrects nobody properly: it spreads the astigmatic blur evenly instead of removing it.
Two properties make it useful. It does not depend on the axis, so it can be computed from a prescription with the axis missing. And it is invariant under transposition: the plus-cylinder and minus-cylinder forms of one refraction give the identical equivalent, because adding the cylinder to the sphere and then subtracting half of it back is the same as adding half of it to the original sphere. Both forms are printed above and they must agree exactly. That makes the spherical equivalent the quickest audit of a transposition anybody has: if the two forms give different equivalents, one of them was typed wrong.
What it is for, in practice, is counting. Epidemiology needs one number per eye to say whether a refraction is myopic, and that is where the spherical equivalent earns its place and where it gets dangerous. Cumberland, Bountziouka and Rahi took one set of autorefractions — 1,985 people from the 1958 British birth cohort, measured in their mid-forties — and applied three published myopia definitions to it: a spherical equivalent at or beyond −1.00 D, −0.75 D, or −0.50 D. Mild-myopia prevalence came out at 28%, 36% and 47%. Their own conclusion is that “even small changes in the threshold definition of myopia (±0.25D) can significantly affect the conclusions” of a study, and that the looser thresholds “resulted in a near doubling of prevalence”. A myopia prevalence quoted without its threshold means nothing.
What it throws away is the astigmatism, and that is not a detail. A refraction of 0.00 −4.00 × 90 has one meridian in perfect focus and one 4.00 D myopic; its spherical equivalent is −2.00 D, the same as a plain −2.00 sphere, and the two eyes see nothing alike. Anything that depends on the cylinder rather than the average — a toric lens, an axis, the corneal contribution to the astigmatism, a vertex compensation, which must be done per meridian at high powers — needs the full prescription. The spherical equivalent is a summary statistic, and the sign of the cylinder is the one thing that must not be dropped on the way in: entering a −2.00 cylinder as 2.00 moves the answer by the whole 2.00 D, in the wrong direction.
Frequently asked questions
What is the formula for spherical equivalent?
Sphere plus half the cylinder. For −2.00 −1.00 the spherical equivalent is −2.00 + (−0.50) = −2.50 D. The axis does not enter the calculation, but the cylinder’s sign does.
Why half the cylinder and not the whole cylinder?
Because the spherical equivalent is the mean of the two principal meridian powers, which are the sphere and the sphere plus the cylinder. Their mean is the sphere plus half the cylinder. Optically it is the power that puts the circle of least confusion on the retina, midway between the two focal lines.
Does the spherical equivalent change if I transpose the prescription?
No, and that is its most useful property. −2.00 −1.00 × 180 and −3.00 +1.00 × 90 are the same lens and both give −2.50 D, because adding the cylinder to the sphere and then taking half of the flipped cylinder back off returns the same number. If the two forms of a prescription give different equivalents, the transposition is wrong.
Can I wear the spherical equivalent instead of the cylinder?
It is not a prescription and this page does not suggest one. Optically the spherical equivalent spreads the astigmatic blur symmetrically rather than removing it, which is why it is used as a summary and as a starting sphere during refraction rather than as a finished correction. A refraction of 0.00 −4.00 × 90 and a plain −2.00 sphere share a spherical equivalent and are not interchangeable.
What counts as myopia in spherical equivalent terms?
It depends on who is counting, and by more than people expect. Published definitions use −0.50 D, −0.75 D and −1.00 D. Applied to one cohort of 1,985 autorefractions those three gave mild-myopia prevalences of 47%, 36% and 28% respectively, with emmetropia falling from 50% to 31%. Thresholds for moderate-to-high myopia (−3.00 D) and hypermetropia (+1.00 D) are more consistent.
Related calculators
References
- Open Exam Prep. Vertex Distance, Sagittal Depth and Spherical Equivalent, Certified Lens Register study guide. open-exam-prep.com. Accessed 9 October 2026.
- Cumberland PM, Bountziouka V, Rahi JS. Impact of varying the definition of myopia on estimates of prevalence and associations with risk factors. Br J Ophthalmol. 2018;102(10):1407-12.
- 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/