Prism Calculator: Combine and Resolve Prism
This prism calculator works in two directions. Compose turns a horizontal and a vertical prism into one resultant prism with its base angle; Resolve breaks a resultant prism at an angle back into its horizontal and vertical parts. Use it to check oblique prism on a prescription or a lensmeter reading, and to practice prism math for the ABO exam.
Prism Composition / Resolution
Combine horizontal + vertical prism into a resultant, or break a resultant prism into its components.
Formula
R = √(H² + V²); θ = atan2(V, H). Reverse: H = R·cos(θ), V = R·sin(θ).
Formulas
Compose: R = √(H² + V²). The base angle is θ = arctan(V ÷ H), placed in the quadrant set by the two base directions. Resolve: H = R × cos θ and V = R × sin θ.
Angle convention: the calculator measures θ counterclockwise from base out: 0° = base out, 90° = base up, 180° = base in, 270° = base down, for either eye. In the 360° notation written on prescriptions (0° at the examiner's right as you face the patient), that matches the left eye. For a right eye, convert both ways with 180° minus the angle, adding 360° if the result is negative: enter 180 minus the written angle in Resolve, and write 180 minus the Compose result on the prescription (217° becomes 323°).
Worked examples
Compose: 2.00Δ BI with 1.50Δ BD. R = √(2.00² + 1.50²) = 2.50Δ, and arctan(1.50 ÷ 2.00) = 36.9°, measured down from base in. The calculator shows 2.50Δ at 217°, which is how it is written for a left lens; on a right lens the same prism is written base 323°.
Resolve, left eye: OS 4.00Δ base 30°. H = 4.00 × cos 30° = 3.46Δ and V = 4.00 × sin 30° = 2.00Δ, so 3.46Δ BO and 2.00Δ BU.
Resolve, right eye: OD 4.00Δ base 150°. Enter 180 − 150 = 30° and the calculator returns the same 3.46Δ BO and 2.00Δ BU, because on a right lens 150° lies between base up (90°) and base out (180°).
Splitting prism between the eyes
The calculator does not split prism, but the rules are short. Unless the prescriber says otherwise, horizontal prism is divided equally with the same base in both eyes: 4.00Δ BO becomes 2.00Δ BO OD and 2.00Δ BO OS. Vertical prism is divided with opposite bases: 3.00Δ BU OD becomes 1.50Δ BU OD and 1.50Δ BD OS. The effect between the eyes stays the same, and the two lenses end up closer in thickness.
When to use it and common mistakes
Use Compose when prism is given in two directions and you need one value, for example to verify an oblique prism on the lensmeter or to order a single resultant prism. Use Resolve when prism is written as one amount and an angle.
Common mistakes: adding H and V directly (2.00 + 1.50 is 3.50, not the 2.50Δ resultant); forgetting that base in and base out sit on opposite sides for the two eyes in 360° notation; and treating the base angle like a cylinder axis. Prism angles run from 0° to 360°, cylinder axes from 1° to 180°.
Prism on the ABO exam
Know the 3-4-5 pattern: 3Δ and 4Δ make 5Δ, and 1.5Δ and 2Δ make 2.5Δ. It lets you check a resultant without a calculator. Prism amounts often come from Prentice's rule, so practice the two together. Review the Prism study guide, then test yourself with the free ABO practice test.
Frequently asked questions
How do you calculate resultant prism?
Square the horizontal and vertical prism, add them, and take the square root: R = √(H² + V²). The base lies between the two base directions, at arctan(V ÷ H) from the horizontal. For example, 3Δ BO with 4Δ BU gives 5Δ, about 53° from base out toward base up.
How is oblique prism written on a prescription?
Either as two parts, such as 2.00Δ BI and 1.50Δ BD, or as one amount with a base angle in 360° notation. Because 0° is at the examiner's right in that notation, the same 2.50Δ prism is written about base 217° on a left lens but about base 323° on a right lens.
Should prescribed prism be split between both eyes?
Usually, unless the prescriber specifies otherwise. Splitting keeps the two lenses closer in thickness and weight. Horizontal prism keeps the same base in each eye; vertical prism takes opposite bases, half base up in one eye and half base down in the other.
Is a prism diopter the same as a degree?
No. One prism diopter (1Δ) displaces an image 1 cm at a distance of 1 m. Prism amounts are given in prism diopters, while base directions in 360° notation are given in degrees.
Related tools and study guides
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