Spherical equivalent calculator

What the spherical equivalent tells you

A glasses prescription with astigmatism has three numbers: a sphere, a cylinder and an axis. The spherical equivalent (SE) folds them into one: the single sphere that would put the blur of an astigmatic eye at the middle of its focal interval, on the retina. Researchers use it to summarize refractive error in one column, and clinicians use it as a quick measure of how nearsighted or farsighted an eye is overall, ignoring the astigmatism.

The formulas, variable by variable

• Spherical equivalent: SE = S + C ÷ 2, where S is the sphere and C the cylinder, both in diopters (D) and both with their signs.
• Transposition: new sphere S′ = S + C, new cylinder C′ = −C, new axis = old axis ± 90°. This switches a prescription between plus-cylinder and minus-cylinder form without changing the lens.
• Power cross: along the axis the power is S; at right angles to it the power is S + C. The AAO course draws these two numbers on a cross, which is the easiest way to check a transposition.

Worked example

The AAO’s optics course gives −8.00 +4.00 × 045. The sphere and cylinder add up to −4.00, the cylinder flips to −4.00 and the axis 45 moves to 135, so the same lens in minus-cylinder form is −4.00 −4.00 × 135. The equivalent is −8.00 + (+4.00 ÷ 2) = −6.00 D, and the minus form agrees: −4.00 + (−4.00 ÷ 2) = −6.00 D. The power cross shows −8.00 D along 45° and −4.00 D along 135°. A second pair from the same course, +1.00 +1.00 × 090, transposes to +2.00 −1.00 × 180, with an equivalent of +1.50 D.

Reading the numbers on a prescription

SPH is the sphere: minus for nearsightedness, plus for farsightedness. CYL is the cylinder that corrects astigmatism, and AXIS is the direction of that cylinder in degrees from 1 to 180. “DS” after a single number means a sphere with no cylinder, and “plano” means zero sphere. Written out, −2.50 −1.50 × 175 reads sphere −2.50, cylinder −1.50, axis 175. Its other form is −4.00 +1.50 × 085: the sphere becomes −4.00, the cylinder +1.50, and the axis moves from 175 to 85. Both write the same lens, and both have an equivalent of −3.25 D.

Checking a transposition by hand with the power cross

Draw two lines, one along the axis and one at right angles. Put the sphere on the axis line and sphere plus cylinder on the other. For +1.00 +1.00 × 090 the axis line gets +1.00 and the other line, 180°, gets +2.00. To write it with a minus cylinder, take the larger plus value, +2.00, as the sphere; the cylinder is the other line minus that sphere, −1.00, and its axis is the line that carries that sphere, 180°. (In either form the power along the axis equals the sphere.) That reproduces +2.00 −1.00 × 180. If your hand check and the calculator disagree, look first for a missed sign.

Limits and common mistakes

• The equivalent hides the astigmatism. −4.00 −4.00 × 135 has an equivalent of −6.00 D, yet no single sphere of that power would give the eye clear sight: one meridian needs −8.00 D and the other −4.00 D. Larger cylinders leave larger blur even when the equivalent is right.
• Forgetting to change the axis. Adding the cylinder to the sphere and flipping its sign is not enough; without the 90-degree axis shift the new prescription describes a different lens.
• Mixing up the axis range. Prescriptions use 1 to 180. An axis of 0 is written 180, and an axis such as 200 is an input error rather than a valid value.
• Treating SE as a purchase order. Spectacle lenses, toric contact lenses and surgery plans are specified with the full sphere, cylinder and axis; vertex distance and lens design also matter. Use the vertex distance calculator when changing between glasses and contact lenses.

Common questions

How do you calculate spherical equivalent?

Add half of the cylinder to the sphere: SE = sphere + cylinder ÷ 2. For −2.50 −1.50 × 175 that is −2.50 + (−1.50 ÷ 2) = −3.25 D. The sign of the cylinder matters, so use the number exactly as written on the prescription, plus or minus.

Does the spherical equivalent change when I switch between plus and minus cylinder?

No. The two forms describe the same lens, so the equivalent is identical. The American Academy of Ophthalmology’s optics course shows +1.00 +1.00 × 090 and +2.00 −1.00 × 180 side by side, both with an equivalent of +1.50 D. If your two numbers differ, one of the transposition steps was done wrong.

How do I convert a plus cylinder prescription to minus cylinder?

Three steps: add the cylinder to the sphere to get the new sphere, change the sign of the cylinder, and move the axis by 90 degrees (add 90 if the axis is 90 or less, subtract 90 if it is more). Axis 0 is written as 180. The calculator does all three and also lists the power along each principal meridian so you can check the result.

Which form should I use, plus or minus cylinder?

Either is correct. Eye surgeons commonly write plus cylinder and optometrists commonly write minus cylinder, but practice differs between clinics and countries. The lens that gets made is the same, so a lab or an optician can work from whichever form you hold.

Can I use the spherical equivalent as a glasses prescription?

No. The equivalent is a single-number summary of where the blur circle lies; it drops the astigmatism correction. Many different sphere and cylinder pairs share one equivalent, so it cannot be turned back into one specific lens. Use it to compare, summarize or sort prescriptions, not to order lenses.

What is the axis of a cylinder?

It is the meridian along which the cylinder has no power. Along the axis the lens power is just the sphere, and 90 degrees away it is sphere plus cylinder. That is why the transposed prescription needs an axis 90 degrees from the original.

Related pages: the visual acuity converter for chart notations, the reading add calculator for near prescriptions and the lens thickness calculator for how power turns into glass.

For general information only — not medical advice. Conversions here are educational and are not a prescription; a prescription must come from an eye-care professional. Source: American Academy of Ophthalmology, Basic and Clinical Science Course, Basic Optics, chapter 15, “Astigmatic Refractive Error: The Power Cross” (the worked power-cross examples and the formula SE = sphere + cylinder ÷ 2). How we check our calculators.