# How do sphere and cylinder measurements combine in eyeglass prescriptions?

For eyeglasses, if a cylinder prescription were combined into a single spherical prescription, how would it be weighted?

As an example, suppose the prescription is S=-3.0, C=+.5×90°.

If only spherical lenses were available, would it be:

• -3.0 — ignore the cylinder value
• -2.5 — add the cylinder value
• -2.75 — add half the cylinder value

or something else.

I assume the "90°" is irrelevant, and strongly suspect the "add half" case is correct or close, but would like to know for sure.

Wikipedia's eyeglass prescription, cylinder component section, says:

"There are two different conventions for indicating the amount of cylinder: "plus cylinder notation" and "minus cylinder notation". In the former, the cylinder power is a number of diopters more convergent than the sphere power. That means the spherical power describes the most divergent meridian and the cylindrical component describes the most convergent. In the minus cylinder notation, the cylinder power is a number of diopters more divergent than the sphere component.

• Add the sphere and cylinder numbers together to produce the converted sphere
• Invert the sign of cylinder value
• Add 90° to axis value, and if the new axis value exceeds 180°, subtract 180° from the result

For example, a lens with a vertical power of -3.75 and a horizontal power of -2.25 could be specified as either -2.25 -1.50 x 180 or -3.75 +1.50 x 090.".

You have "S=-3.0, C=+.5×90°", so:

1. -3.0 + .5 = -2.5
2. .5 * -1 = -.5
3. 90° + 90° = 180°

If only spherical lenses were available ...

Source: What is a Spherical Equivalent?:

The axis is removed

• The axis is not part of the calculation for spherical equivalent at all. It completely disappears.

The cylinder is divided by 2

• The cylinder power is only present in a particular direction. In other words, it is only present in half the lens. When attempting to combine the cylinder with the sphere, your doctor must take this into account by only taking half of the cylinder.
• Because cylinder powers only come in steps of -0.25, it is possible that when it is divided by 2, that the number does not end in a multiple of -0.25. That is where your eye doctor’s professional judgement comes into play to decide which closest multiple of -0.25 will be the most appropriate for you based on many different considerations.

The sphere and the 1/2 cylinder are combined

• The sphere and the cylinder are then added together to give you the equivalent sphere.

Answer: -2.5 + (-.5 / 2) = -2.75

The choice "add half the cylinder value" is correct.