ABO Exam Formulas: A Plain-Language Cheat Sheet

Most ABO math comes down to a handful of formulas: power = 1 ÷ focal length, vergence (V = U + P), transposition, spherical equivalent, Prentice’s rule (prism = cm × D), decentration and minimum blank size, and vertex compensation. Ophthalmic Optics is the largest domain on ABO-NCLE’s published outline, so these are worth practicing until they are automatic. ABO-NCLE builds a scientific calculator into the exam screen.

Last checked Nov 2, 2026. Confirm fees and dates with the certifying body. optical.education is independent and is not affiliated with or endorsed by ABO-NCLE. Disclaimer.

Why the math matters

On ABO-NCLE’s published outline, Ophthalmic Optics has 25 of the 100 specified questions, more than any other domain. You also need math for decentration and blank size in dispensing. Each formula below has a short worked example. Distances in formulas are in meters unless it says otherwise.

Lens power and vergence

• Power: P = 1 ÷ f (f in meters). A lens with a 25 cm focal length is 1 ÷ 0.25 = 4.00 D.
• Vergence: V = U + P. U = 1 ÷ object distance, which is negative for a real object in front of the lens. Object at 50 cm: U = −2.00 D. With a +5.00 D lens, V = +3.00 D, so the image forms 1 ÷ 3 = 33 cm behind the lens.
• Accommodation: demand = 1 ÷ near distance. Reading at 40 cm needs 2.50 D of focusing (or add).

Try it: vergence calculator · accommodation calculator.

Transposition and spherical equivalent

• Transposition: new sphere = sphere + cylinder; flip the cylinder’s sign; axis ± 90. Example: +1.00 −2.00 × 180 becomes −1.00 +2.00 × 090.
• Spherical equivalent: SE = sphere + (cylinder ÷ 2). Example: −2.00 −1.00 × 090 has an SE of −2.50 D.
• Power in a meridian (power cross): the sphere power is on the axis meridian; sphere + cylinder is 90 degrees away.

Try it: transposition calculator · spherical equivalent calculator · transposition guide.

Prism and Prentice’s rule

• Prentice’s rule: prism (Δ) = c × D, where c is the distance from the optical center in centimeters. Example: 3 mm off center in a −4.00 D lens: 0.3 × 4 = 1.2Δ.
• Base direction: plus lenses act like two prisms base to base, so the base points toward the optical center. Minus lenses are apex to apex, so the base points away from it.
• Combining prism: if both eyes have prism in the same direction (both base in, or both base up), the effects cancel; opposite directions add. For horizontal and vertical prism in one lens, the resultant = √(H² + V²).

Try it: Prentice’s rule calculator · Prentice’s rule guide.

Decentration and minimum blank size

• Frame PD: A (eye size) + DBL (bridge). A 52-18 frame has a frame PD of 70 mm.
• Decentration per lens: (frame PD − patient PD) ÷ 2. With a 64 mm PD in that frame: (70 − 64) ÷ 2 = 3 mm in per lens. Use monocular PDs when you have them: decentration = (frame PD ÷ 2) − monocular PD.
• Minimum blank size: ED + (2 × decentration per lens) + 2 mm. With a 56 mm ED: 56 + 6 + 2 = 64 mm.

Try it: decentration and blank size calculator.

Vertex distance

Compensated power: F_new = F ÷ (1 − d × F), where d is how far the lens moves toward the eye, in meters (use a negative d if it moves away). Example: a −8.00 D lens moved 12 mm closer (to the cornea): −8 ÷ (1 + 0.096) = −7.30 D. A plus lens moved closer needs more plus; a minus lens moved closer needs less minus. It starts to matter above about ±4.00 D.

Try it: vertex distance calculator.

Surfaces, index and the lens clock

• Index of refraction: n = speed of light in air ÷ speed in the material. Higher index means a thinner lens for the same power.
• Surface power: F = (n − 1) ÷ r (r in meters). Nominal lens power ≈ front curve + back curve.
• Lens clock correction: lens clocks are usually calibrated for index 1.53. True power = clock reading × (n − 1) ÷ 0.53.
• Sag (approximate): s ≈ y² ÷ (2r), where y is half the lens diameter. Bigger lenses and steeper curves mean more sag and more edge or center thickness.

Compare materials: lens materials comparison.

Exam-day math tips

• Convert units first: millimeters to centimeters for Prentice, centimeters to meters for power and vergence.
• Write down the sign of every power before you start. Most wrong answers on these items are sign errors.
• Estimate before you calculate, so you can spot an answer that is ten times too big.
• At 125 questions in 2 hours (ABO-NCLE’s format), you have under a minute per question, so practice until each formula takes seconds.