Snell’s Law Calculatorn₁ sin θ₁ = n₂ sin θ₂
Snell’s law describes how light bends at a boundary: n₁ sin θ₁ = n₂ sin θ₂, with angles measured from the normal. Going into a denser medium, light bends towards the normal. Going into a less dense one, it bends away, and beyond the critical angle it is totally reflected. Enter the two refractive indices and an angle to get the refraction angle, the critical angle and a scaled ray diagram.
Formula verified against OpenStax — University Physics, Volume 3 · HyperPhysics, Georgia State University
By Manzoor Ahmad · Pharmacist & calculator developerReviewed & updated
Two media
Air 1.0003 · water 1.333 · glass 1.50–1.52 · diamond 2.417
Step-by-step working
What is snell’s law?
Snell’s law describes refraction, the bending of light as it passes from one transparent medium into another where it travels at a different speed. Each medium is described by its refractive index n.
It explains why pools look shallower than they are, how lenses and eyeglasses focus light, why diamonds sparkle, and how fibre-optic cables carry internet traffic by total internal reflection.
Quick answers
How do you calculate the angle of refraction?
Use θ₂ = sin⁻¹(n₁ sin θ₁ ÷ n₂). Light from air hitting glass (n = 1.5) at 40° refracts to 25.4°.
What is the critical angle?
The angle of incidence beyond which light can’t leave a denser medium: θc = sin⁻¹(n₂ ÷ n₁). For water to air it is 48.6°; for diamond, 24.4°.
How do I find the refractive index from angles?
n₂ = n₁ sin θ₁ ÷ sin θ₂. A laser entering a block at 45° and refracting to 28.1° indicates n ≈ 1.50, typical glass. Use the Find n₂ tab.
Why does a straw look bent in water?
Light from the submerged part refracts away from the normal as it leaves the water, so your brain traces it back to a shallower position. A converging lens uses the same bending deliberately.
Snell’s law formula
Total internal reflection happens only going from a higher to a lower index, when θ₁ > θc.
| Symbol | Meaning | SI unit |
|---|---|---|
| n₁, n₂ | Refractive indices of the two media | — |
| θ₁ | Angle of incidence (from the normal) | ° |
| θ₂ | Angle of refraction | ° |
| θc | Critical angle | ° |
| v | Speed of light in the medium | m/s |
How to use Snell’s law
- Draw the normal at the point where the ray hits the boundary.
- Measure θ₁ from the normal, not from the surface.
- Compute sin θ₂ = n₁ sin θ₁ ÷ n₂.
- If the value is above 1, the light is totally internally reflected.
Worked example: light entering water at 30°
A ray in air (n = 1.0003) hits water (n = 1.333) at 30° from the normal.
θ₂ = 22.04°
Going the other way, from water into air, any ray steeper than the critical angle sin⁻¹(1.0003 ÷ 1.333) = 48.6° is trapped by total internal reflection. For crown glass (n = 1.5) it is 41.8°.
Refractive index of common materials
At the sodium D line (589 nm), approximate.
| Material | n | Critical angle to air |
|---|---|---|
| Air (STP) | 1.0003 | — |
| Ice | 1.31 | 49.8° |
| Water | 1.333 | 48.6° |
| Acrylic (PMMA) | 1.49 | 42.2° |
| Crown glass | 1.52 | 41.1° |
| Flint glass | 1.62 | 38.1° |
| Sapphire | 1.77 | 34.4° |
| Diamond | 2.417 | 24.4° |
Common mistakes to avoid
- Measuring angles from the surface. Snell’s law uses angles from the normal.
- Calculator in radians. sin 30 rad ≠ sin 30°.
- Expecting refraction past the critical angle. Above θc there is no refracted ray at all.
Manzoor Ahmad
Pharmacist turned Web Developer & Calculator Engineer · 13+ years in web developmentI’m a pharmacist by training who turned into a full-time tech geek, and for 13+ years I’ve designed and coded web tools. My pharmacy degree included physical pharmacy and pharmaceutics — states of matter, heat, flow and reaction kinetics — so the physics behind these tools is familiar ground, not a new hobby.
I’m not a physicist, so I don’t ask you to take my word for it: every formula on this page is traced to an open university textbook or NIST reference data, and the calculator is tested against published worked examples before it goes live. Spot an error? Tell me and it will be fixed and dated on the next update.
Snell’s Law Calculator: frequently asked questions
A rule relating the angles of incidence and refraction at a boundary: n₁ sin θ₁ = n₂ sin θ₂.
n = c ÷ v, the ratio of the speed of light in a vacuum to its speed in the material.
When light travels from a higher-index to a lower-index medium at an angle greater than the critical angle.
No. Frequency stays the same; speed and wavelength both drop by the factor n.
The refractive index varies slightly with wavelength (dispersion), so blue bends more than red, which is how prisms make rainbows.
References & sources
- OpenStax — University Physics, Volume 3 — Ch. 1.3–1.4, Refraction and Total Internal Reflection
- HyperPhysics, Georgia State University — Snell’s law and refractive indices
- RefractiveIndex.INFO database — measured refractive indices
Disclaimer: This calculator is an educational tool built on idealised textbook models. It is not a substitute for professional engineering or safety calculations.