Angle of Refraction Calculator — Snell's Law
Apply Snell's law (n₁·sin θ₁ = n₂·sin θ₂) to find the angle of refraction when light crosses the boundary between two media. Choose from common material presets or enter a custom refractive index.
°
Second medium
Angle the transmitted ray makes with the normal in the second medium (Snell's law)
- 1
sin(θ₁)
sin(30° × π ÷ 180) = 0.5 - 2
Snell ratio n₁ · sin θ₁ ÷ n₂
1 × 0.5 ÷ 1.5 = 0.333333Snell's law: n₁ sin θ₁ = n₂ sin θ₂. - 3
Angle of refraction θ₂
arcsin(0.333333) × (180 ÷ π) = 19.47
How does this calculator work?
Snell's law: n₁·sin θ₁ = n₂·sin θ₂, so θ₂ = arcsin(n₁·sin θ₁/n₂). For air-to-glass (n₂=1.5) at 30° incidence, θ₂ ≈ 19.5°. If n₁ > n₂ and θ₁ exceeds the critical angle arcsin(n₂/n₁), total internal reflection occurs and no refracted ray exists. Refractive indices here are for visible light near 589 nm.
Formula
How this is calculated
When a ray crosses the boundary between two transparent media it changes speed and bends — this is refraction. Snell's law connects the incidence angle θ₁ and refraction angle θ₂ through the refractive indices: n₁·sin θ₁ = n₂·sin θ₂. Both angles are measured from the surface normal, not from the surface itself. The refractive index n = c/v is the ratio of the speed of light in vacuum (c ≈ 3 × 10⁸ m/s) to the phase speed in the medium.
If the ray enters a denser medium (n₂ > n₁), it slows down and bends toward the normal: θ₂ < θ₁. Going from a denser to a less dense medium it bends away (θ₂ > θ₁). When n₁ > n₂ and θ₁ reaches the critical angle θ_c = arcsin(n₂/n₁), the refracted ray lies along the surface (θ₂ = 90°). Beyond θ_c, the ratio n₁·sin θ₁/n₂ exceeds 1 and no real solution exists — total internal reflection (TIR) occurs. TIR is the basis of optical fibres and explains why diamonds and cut gemstones sparkle so brilliantly (critical angle ≈ 24.4° for diamond).
This calculator uses a single fixed n per medium; real materials are dispersive (n varies with wavelength), so white light separates into colours at a prism or water droplet. The tabulated n values are for visible light near 589 nm (sodium D-line).
Frequently asked questions
Light slows down in denser media. When the wavefront hits the boundary at an angle, one side enters and slows first while the other still travels at the original speed. This rotation of the wavefront bends the ray toward the normal — the same effect as a wheel veering when one side hits rough ground.
TIR occurs when light travels from a denser (high-n) medium into a less dense (low-n) one at an angle greater than the critical angle θ_c = arcsin(n₂/n₁). At that point Snell's law would require sin θ₂ > 1, which is impossible — so all light reflects back. Optical fibres, diamonds, and certain microscopy techniques exploit TIR.
Most transparent materials have a refractive index that varies with wavelength — red light has a lower n and bends less; violet light has a higher n and bends more. This is why white light splits into a rainbow in a glass prism or raindrop. This calculator uses a single n per medium; for dispersion you would need wavelength-dependent Sellmeier coefficients.
Also known as
TG we-Calculate Editorial Team. (2026). Angle of Refraction Calculator — Snell's Law [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/angle-of-refraction-calculator
TG we-Calculate Editorial Team. "Angle of Refraction Calculator — Snell's Law." TG we-Calculate. 2026. https://we-calculate.com/calculator/angle-of-refraction-calculator.
TG we-Calculate Editorial Team, "Angle of Refraction Calculator — Snell's Law," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/angle-of-refraction-calculator
@misc{wecalculate_angle_of_refraction_calculator, title = {Angle of Refraction Calculator — Snell's Law}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/angle-of-refraction-calculator}}, year = {2026}, note = {TG we-Calculate} }
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