Snell’s Law & Refraction Calculator
Work out how much light bends at a boundary. Solve n₁ sin θ₁ = n₂ sin θ₂ for either angle or either refractive index, find the critical angle for total internal reflection, and see the rays drawn at the surface.
How to Use
- Pick what to find: the refracted angle, the incident angle, either refractive index, or the critical angle.
- Choose the two media (vacuum, air, water, acrylic, crown glass or diamond) or type any refractive index. Medium 1 is where the light starts.
- Enter the angles in degrees or radians, measured from the normal (the line at right angles to the surface), not from the surface itself.
- Read the answer and the ray diagram. If the light cannot get out, the calculator says so: that is total internal reflection.
- Press a preset to load an example, and check Show Work for each step.
Worked Example
Air into water at 45°. sin θ₂ = n₁ sin θ₁ ÷ n₂ = 1.000293 × 0.707107 ÷ 1.333 = 0.530618, so θ₂ = arcsin 0.530618 = 32.05°. Going into the denser water, the ray bends towards the normal by 12.95°.
Water into air: the critical angle. Going the other way the ray bends away from the normal, and at sin θc = 1.000293 ÷ 1.333 = 0.750407 it lies flat along the surface: θc = 48.63°. At 60°, sin θ₂ would be 1.333 × 0.866025 ÷ 1.000293 = 1.154, which no angle has, so the light is totally internally reflected.
The common mistake: measuring from the surface. A ray meets water at 30° to the surface. Its angle from the normal is 90° − 30° = 60°, and it refracts to 40.53° from the normal. Using 30° directly gives 22.04°, a different ray altogether.
Show Work
Formulas
Ibn Sahl, Snellius and Descartes
Ptolemy measured tables of refraction angles from air into water and glass in the 2nd century AD, but found no rule. The first correct statement of the law is in a treatise on burning mirrors and lenses written in Baghdad around 984 by Ibn Sahl, who used it to design lenses that focus light without aberration. In the West, the Dutch mathematician Willebrord Snellius found the same law in 1621 but never published it; René Descartes published it in La Dioptrique in 1637, which is why the French still call it the Snell–Descartes law. Pierre de Fermat derived it in 1662 from the principle that light takes the quickest path.
David Brewster described the angle of zero reflection for polarised light in 1815, and Augustin-Jean Fresnel worked out how much light is reflected at any angle in the 1820s. Refractive indices here are for yellow sodium light (589 nm), from the CRC Handbook of Chemistry and Physics; they vary slightly with colour, which is why a prism splits white light.
About This Calculator
This calculator solves Snell’s law for the refracted angle, the incident angle or either refractive index, and finds the critical angle. It detects total internal reflection instead of returning an error, and adds what happens at the same surface: the share of light reflected (Fresnel), Brewster’s angle and the speed of light in each medium. The ray diagram shows the incident, reflected and refracted rays with their angles.
Everything runs in your browser; nothing you enter is sent anywhere.
Related tools: Thin Lens & Mirror Calculator, Doppler Effect & Wave Calculator, and Frequency & Wavelength Calculator.
Frequently Asked Questions
What is Snell’s law?
When light crosses from one material into another, n₁ sin θ₁ = n₂ sin θ₂, where n is each material’s refractive index and θ is measured from the normal. Going from air (n = 1.000293) into water (n = 1.333) at 45°, sin θ₂ = 1.000293 × sin 45° ÷ 1.333 = 0.5306, so the ray bends to 32.05°.
What is the critical angle?
It is the angle of incidence at which light leaving a denser medium is refracted along the surface (θ₂ = 90°): sin θc = n₂ ÷ n₁. From water into air it is arcsin(1.000293 ÷ 1.333) = 48.63°; from crown glass (1.52) into air it is 41.15°; from diamond (2.417) it is only 24.45°.
What is total internal reflection?
Beyond the critical angle, n₁ sin θ₁ ÷ n₂ comes out above 1, no angle has that sine, and all the light reflects back. Light in water hitting the surface at 60° gives 1.333 × sin 60° ÷ 1.000293 = 1.154, so none escapes. Optical fibres and the sparkle of a cut diamond both rely on it.
Why must the angle be measured from the normal?
Snell’s law is written for angles from the normal. A ray hitting water at 30° to the surface is at 60° from the normal and bends to 40.53°. Putting 30° into the formula gives 22.04°, which is wrong by more than 18°.
How much light reflects off glass?
At normal incidence the reflected fraction is ((n₂ − n₁) ÷ (n₂ + n₁))². For air to crown glass that is (0.519707 ÷ 2.520293)² = 4.25%, so an uncoated window loses about 4% at each surface. At Brewster’s angle, arctan(1.52 ÷ 1.000293) = 56.65°, light polarised in the plane of incidence is not reflected at all.
How do I use the Snell’s Law & Refraction Calculator?
Simply type your numbers and read the result, which refreshes the instant you change something. There is nothing to submit and nothing to wait for.
Is it free? Does it work without internet?
Yes to both. It is free with no sign-up, and once the page has loaded it keeps working even with no internet.
Where does my data go?
Nowhere — every calculation runs on your own device. Nothing you enter is uploaded, logged, or stored.
Common Use Cases
Physics homework
Light from air into crown glass at 30° bends to 19.21°, and the speed of light in that glass is 299,792 ÷ 1.52 = 197,232 km/s.
Finding an unknown index
Measure a ray going in at 40° and leaving at 25° below the surface of a glass block: n = 1.000293 × sin 40° ÷ sin 25° = 1.521, which is crown glass.
Fibres and light pipes
Light inside acrylic (1.49) stays trapped as long as it meets the wall at more than arcsin(1.000293 ÷ 1.49) = 42.17° from the normal, the critical angle.
Underwater views
A diver looking up sees the whole sky squeezed into a circle 2 × 48.63° = 97.3° across (Snell’s window); outside it the surface acts as a mirror.
Gem cutting
Diamond’s critical angle of 24.45° is why its facets trap and return so much light compared with glass at 41.15°.
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