Snell’s Law Calculator
The angle light bends to as it crosses from one material into another, n₁ sin θ₁ = n₂ sin θ₂, or either index or the incident angle, with the critical angle.
Result
Enter refractive index the light comes from, n₁ and refractive index the light goes into, n₂ to see the result.
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Results come from the published equations applied to the values you enter. They are not electrical design, conductor sizing, laboratory safety or medical guidance. Weather indices describe a standard person in standard conditions and are not a forecast. Where codes apply, local rules vary and must be checked. Read the full disclaimer
How to use the Snell’s Law Calculator
- Choose what to find, usually the angle of refraction.
- Enter the refractive index of the material the light comes from and the one it goes into, from a table or the material’s data sheet.
- Enter the angle, measured from the normal, the line square to the surface, and read the other angle and the critical angle.
The formulas
n₁ × sin θ₁ = n₂ × sin θ₂θ₂ = sin⁻¹(n₁ sin θ₁ ÷ n₂)n₂ = n₁ sin θ₁ ÷ sin θ₂critical angle θc = sin⁻¹(n₂ ÷ n₁), when n₁ > n₂Angles are measured from the normal, not from the surface. A refractive index is the speed of light in a vacuum divided by its speed in the material.
Limits
The boundary is taken as flat and both materials as uniform. The page does not work out how much light is reflected at the surface below the critical angle, and it uses one index per material, whereas real indices vary a little with the colour of the light.
With your numbers
- n₁ sin θ₁ = 1 × sin 30 ° = 0.5
- n₂ sin θ₂ = 2.419 × sin 11.9289 ° = 0.5
Quick reference
Indices are for one wavelength; they change slightly with colour, which is how a prism splits white light.
| Example | Result |
|---|---|
| Air 1.00 into diamond 2.419 at 30° | Angle of refraction: 11.93 ° |
| n₂ bending 30° to 22° from air at 1.00 | Refractive index of the second medium: 1.334734 |
| Glass 1.5 out into air 1.0 at 30° | Angle of refraction: 48.59 ° |
| Polystyrene 1.49 into air 1.00 at 42°, just under the critical angle | Angle of refraction: 85.56 ° |
| Flint glass 1.66 into crown glass 1.52 at 45° | Angle of refraction: 50.56 ° |
| Incidence for 11.93° in diamond from air | Angle of incidence: 30 ° |
Common questions about the Snell’s Law Calculator
- What is Snell’s law?
- n₁ sin θ₁ = n₂ sin θ₂. Light crossing into a material with a higher refractive index bends towards the normal, and into a lower one bends away. OpenStax University Physics Volume 3, example 1.3, sends light from air, n = 1.00, into diamond, n = 2.419, at 30.0°: it bends to about 11.9°.
- How do I find a refractive index from two angles?
- Rearrange: n₂ = n₁ sin θ₁ ÷ sin θ₂. OpenStax example 1.2 has light from air at 30.0° bending to 22.0°, which gives n₂ ≈ 1.33, the index of water.
- What is the critical angle?
- Going from a denser medium into a less dense one, the refracted ray bends away from the normal, and at the critical angle it runs along the surface. Beyond that no light gets out: it is all reflected back, which is total internal reflection. θc = sin⁻¹(n₂ ÷ n₁); for polystyrene, n = 1.49, into air it is about 42.2°.
- Why is there no answer for some angles?
- Past the critical angle Snell’s law would need a sine greater than 1, so there is no refracted ray. The page says total internal reflection instead of printing a number. It is how optical fibres keep light inside them and why a diamond, with a critical angle of about 24.4°, sparkles.
- Where do I get refractive indices?
- From a reference table or the maker’s data for the glass, plastic or liquid you have. The page does not guess one, because it depends on the material and slightly on the colour of the light. A vacuum is exactly 1 and air very close to it.