Prism Refraction Angle Calculator

When light passes through a prism, it undergoes refraction, causing the light to change direction based on the geometry and refractive index of the prism material.
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CalculatorHow to use:Select the angle you want to find from the drop down on the "Select angle to find" column in the Input table.
Enter the three known angles.
See the angle value in the Output table.
See input parameter descriptions
Relevant Formulas
Inputs﻿
Output﻿
θA​
20​
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Additional InformationFormulaThe angled denoted above are connected using the formula below.
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θA = θ1 + θE − θD\theta_A\ =\ \theta_1\ +\ \theta_E\ -\ \theta_DθA​ = θ1​ + θE​ − θD​
This formula can be rearranged to find any of the missing parameters given three other values are provided.
ParametersAngle of incidence (﻿θ1​
) - The angle between the normal to the incident surface and Incident ray.
Angle of Deviation (﻿θD​
) - The angle by which the incident light ray is deviated or bent inside the prism. It is the angle between the incident ray and the emergent ray. 
Angle of Emergence (﻿θE​
) - The angle between the normal to the exiting surface and the Emergent ray. 
Prism Refraction Angle (﻿θA​
) - Also known as the Prism Angle and is the angle by which light refracts within the prism.
DerivationIn this section, a detailed layout of the derivation of the formula is shown.
The diagram below will be referred to during the derivation.
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Layout of Prism Angles
We will now break down this prism to find equations to define the individual parameters. Each step is described below:
Find Equation 1 - Analyse Triangle MPQ
Find Equation 2 - Analyse Point P
Find Equation 3 - Analyse Point Q
Find Equation 4 - Analyse Quadrilateral PNQM
Find Equation 5 - Analyse Triangle PNQ
Now we can combine all the equations to solve for the final equation. Shown below are the final steps.
Substitute Equation 2 and 3 into Equation 1 to get Equation 6
Substitute Equation 5 into Equation 6 and get Equation 7
Substitute Equation 4 into Equation 7 and simplify to find the formula
Useful ResourcesA video that clearly explains the Derivation
Find calculators to calculate the Angle of Incidence
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