Angle of Incidence Calculator
This calculator applies Snell's Law to solve angle of incidence, angle of refraction, or refractive index problems in optics. Pick what you want to find from the solve-for menu: angle of incidence (theta1), angle of refraction (theta2), or the refractive index of either medium (n1 or n2). Then enter the refractive indices for medium 1 (the incident medium) and medium 2 (the refracted medium), choosing from presets such as air or vacuum, water, crown glass, borosilicate glass, and ice, or typing a custom value. Enter whichever angle is known, in degrees or radians, and the calculator solves for the rest. Results show the angle of incidence, angle of refraction, angle of reflection (always equal to the angle of incidence) and the critical angle for total internal reflection, alongside a full breakdown of the n1 sin(theta1) = n2 sin(theta2) calculation, the sine values used, the ratio between the two refractive indices, and whether the ray bends toward or away from the normal. It also flags when total internal reflection occurs, such as light travelling from glass or water back into air at too steep an angle. All angles are measured from the normal, the line perpendicular to the surface, not from the surface itself. It is useful for physics study, optics coursework, and checking lens, prism or fibre-optic calculations.
1. Solve For
2. Known Angles
Snell's Law Breakdown
Additional Results
What Is the Angle of Incidence?
The angle of incidence is the angle between an incoming ray (of light, sound, or another wave) and the normal to the surface at the point of contact. The normal is an imaginary line drawn perpendicular to the boundary between two media. The angle is always measured from the normal, not from the surface itself.
When a ray of light travels from one medium into another (for example, from air into water), it changes direction. This bending of light is called refraction. How much the ray bends depends on the refractive indices of both media and the angle at which the ray strikes the boundary.
Snell's Law
The relationship between the angle of incidence and the angle of refraction is described by Snell's Law:
Where:
- n1 is the refractive index of the first (incident) medium
- theta1 is the angle of incidence (measured from the normal)
- n2 is the refractive index of the second (refracted) medium
- theta2 is the angle of refraction (measured from the normal)
This calculator lets you solve for any one of the four values given the other three.
The Law of Reflection
Alongside refraction, some of the incident light is always reflected at the boundary. The law of reflection states that the angle of incidence equals the angle of reflection:
Both angles are measured from the normal, and all three rays (incident, reflected, refracted) lie in the same plane.
Total Internal Reflection and the Critical Angle
When light travels from a denser medium (higher n) to a less dense medium (lower n), there is a maximum angle of incidence beyond which no refraction occurs. All the light is reflected back into the first medium. This is called total internal reflection, and the minimum angle at which it occurs is the critical angle:
Total internal reflection is the principle behind optical fibres, periscopes, and diamond brilliance.
Common Refractive Indices
| Medium | Refractive Index (n) |
|---|---|
| Vacuum | 1.000 (exact) |
| Air (at standard conditions) | 1.000293 |
| Ice | 1.309 |
| Water (at 20 deg C) | 1.333 |
| Ethanol | 1.361 |
| Crown glass | 1.500 |
| Borosilicate glass | 1.523 |
| Diamond | 2.417 |
Worked Example
A ray of light travels from air (n1 = 1.000) into water (n2 = 1.333) at an angle of incidence of 45 degrees. What is the angle of refraction?
Applying Snell's Law:
n1 sin(theta1) = n2 sin(theta2)
1.000 x sin(45 deg) = 1.333 x sin(theta2)
0.7071 = 1.333 x sin(theta2)
sin(theta2) = 0.7071 / 1.333 = 0.5304
theta2 = arcsin(0.5304) = 32.03 deg
The ray bends toward the normal because water is denser than air (n2 > n1). The angle of reflection at the air-water surface is also 45 degrees (law of reflection).
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Sources: Hecht, E. (2017). Optics (5th ed.). Pearson. Snell's Law derivation and refractive index values: NIST Physical Reference Data (physics.nist.gov). Refractive index database values: refractiveindex.info.
This calculator applies the standard Snell's Law formula for monochromatic light at a planar interface. Refractive indices vary with wavelength (dispersion); the values shown are approximate values for visible light at standard conditions. For precise optical engineering work, consult a full spectral data source.