Calculator + drills
Meridian power
The power of an astigmatic lens in any meridian is F = S + C x sin^2(theta - axis). Along the cylinder axis the power is the sphere alone; 90 degrees away it is sphere plus cylinder; 45 degrees off axis it is sphere plus half the cylinder. The dial below presents it the way a lensometer reticle does.
Meridian power calculator and lensometer dial
Enter each eye and pick the meridian to inspect. The dial shows the cylinder axis and the inspected meridian the way a lensometer reticle presents them; the power cross shows both principal meridians.
OD
Power at 90 degrees-3.50 D
F(theta) = S + C x sin^2(theta - axis)
- F(theta) = S + C x sin^2(theta - axis).
- theta - axis = 90 - 180 = -90 degrees; sin^2 = 1.0000.
- F = -2.00 + (-1.50 x 1.0000) = -3.50 D.
Power cross
Sphere meridian at 180 degrees; combined power 90 degrees away at 090.
OS
Power at 90 degrees+1.13 D
F(theta) = S + C x sin^2(theta - axis)
- F(theta) = S + C x sin^2(theta - axis).
- theta - axis = 90 - 45 = 45 degrees; sin^2 = 0.5000.
- F = +1.50 + (-0.75 x 0.5000) = +1.13 D.
Power cross
Sphere meridian at 045 degrees; combined power 90 degrees away at 135.
Practice meridian power
Unlimited generated problems with exact answers and worked steps.
Go deeper
Turning the two line focuses into a written prescription is covered in reading a lensometer. The vertical meridian power drives vertical imbalance and per-eye Prentice prism.
Frequently asked questions
How do you find the power of a lens in a given meridian?
Use F = S + C x sin^2(theta - axis), where theta is the meridian you care about. Along the axis the power is the sphere alone; 90 degrees away it is sphere plus cylinder; 45 degrees away it is sphere plus half the cylinder.
Why does the sine squared term appear?
Cylinder power grows from zero at its axis to full value 90 degrees away, and the growth follows sin^2 of the angle from the axis. It is a smooth transition, not a jump.
What are the principal meridians?
The two meridians 90 degrees apart holding the extreme powers: the sphere power along the axis and sphere plus cylinder perpendicular to it. A lensometer reads these two as its two line focuses.