Snell's Law Calculator
Solve the refraction law n₁·sinθ₁ = n₂·sinθ₂ for any one of the four variables. Detects total internal reflection automatically, computes the critical and Brewster angles, and visualises the ray geometry.
| Material | n (589 nm, 20 °C) | Critical angle vs air | Brewster angle vs air |
|---|
| θ₁ (incidence) | θ₂ (refraction) | sinθ₁ / sinθ₂ | Status |
|---|
| Interface | n₁ → n₂ | Critical angle θc | Brewster angle θB |
|---|
Denser → rarer (n₁ > n₂)
Light bends away from the normal. There is a critical angle θc = arcsin(n₂/n₁) beyond which the ray cannot escape and is completely reflected. This is total internal reflection — the mechanism behind fibre optics, prism binoculars and gemstone sparkle.
Rarer → denser (n₁ < n₂)
Light bends toward the normal. The refracted ray is always on the same side as the incident ray (θ₂ < θ₁), and there is no critical angle — the ray always transmits into the denser medium for any θ₁.
Normal incidence (θ₁ = 0)
When light hits a boundary perpendicular to the surface, it passes straight through without bending, whatever the indices. Snell's law reduces to 0 = 0, so n₂ cannot be determined from normal-incidence data alone.
Brewster's angle
At θB = arctan(n₂/n₁), the reflected beam is perfectly polarised parallel to the surface. For air-to-glass (n₂ = 1.52), θB ≈ 56.7°. Polarising sunglasses exploit this — glare off water and roads is partially polarised at Brewster's angle.
How to Use This Calculator
Snell's law relates the angles of incidence and refraction at the boundary between two transparent media. Enter any three of the four values and the fourth appears instantly.
1. Choose what to solve for
Pick θ₂ to find the refracted ray direction from the two indices and the incidence angle. Pick n₂ to determine an unknown medium from observed angles. Pick θ₁ or n₁ for the reverse problems.
2. Enter the three known values
Type a number, drag the slider, or tap a preset. Indices have their own quick-fill buttons (air, water, glass, diamond). Angles can be entered in degrees, radians or gradians.
3. Watch for TIR
If you solve for θ₂ and the combination n₁ sinθ₁ / n₂ exceeds 1, the ray undergoes total internal reflection. The calculator flags it clearly with an orange badge instead of an error.
4. Check the charts
The Ray diagram draws the boundary, normal, incident ray, refracted ray and reflected ray (dashed) with angles annotated. θ₂ vs θ₁ traces the full Snell curve with the critical angle marked. TIR / transmission zones shows the safe and forbidden incidence ranges.
5. Reference tables
Table 1 lists refractive indices of common materials with their critical and Brewster angles. Table 2 shows refraction across a grid of incidence angles for your current interface. Table 3 (Advanced) lists critical angles for common material pairs.
6. Advanced mode
See sensitivity to each input, plus companion metrics: critical angle, Brewster angle, sinθ₁/sinθ₂ ratio, and the direction of bending (toward or away from the normal).
What This Tool Offers
A complete workbench for refraction problems at a planar interface — the standard tool for optics coursework, lens design, and laboratory data reduction.
Four-way solve
Solve for any one of n₁, θ₁, n₂, θ₂ from the other three. Automatic total internal reflection detection when solving for θ₂.
Three interactive views
Ray diagram with boundary, normal, refracted and reflected rays; the classic θ₂-vs-θ₁ curve with critical angle; and a transmission/TIR zone bar.
Angle unit switching
Incident and refracted angles switch independently between degrees, radians, and gradians.
Reference tables
Refractive indices of common media, refraction across a grid of incidence angles for your interface, and critical angles for common material pairs.
Sensitivity check
See how a +1% change in each input moves the result — useful for understanding measurement uncertainty in prism and refractometer work.
Companion quantities
Critical angle, Brewster's angle, sinθ₁/sinθ₂ ratio, bending direction, and a check for whether TIR is possible at this interface.
What Is Snell's Law?
Snell's law (also called the law of refraction) describes how a ray of light changes direction when it crosses the boundary between two transparent media of different refractive indices. It states that the product of the refractive index and the sine of the angle to the normal is conserved across the boundary: n₁ sinθ₁ = n₂ sinθ₂. The angles are always measured from the normal — an imaginary line perpendicular to the surface at the point of incidence. When light passes from a rarer medium (lower n) into a denser one (higher n), it bends toward the normal; when it passes the other way, it bends away. Beyond a critical angle when leaving a denser medium, the light cannot escape at all and is entirely reflected — the phenomenon of total internal reflection that makes optical fibres, prismatic binoculars, and diamond sparkle possible.
Formula and Variables
Critical angle: θc = arcsin(n₂ / n₁) (defined only when n₁ > n₂)
Brewster angle: θB = arctan(n₂ / n₁)
| Symbol | Meaning | Notes |
|---|---|---|
| n₁ | Refractive index of incident medium | Dimensionless, ≥ 1 for ordinary materials |
| θ₁ | Angle of incidence | Measured from the normal, in [0°, 90°] |
| n₂ | Refractive index of refracting medium | Dimensionless, ≥ 1 |
| θ₂ | Angle of refraction | Measured from the normal, in [0°, 90°] |
| θc | Critical angle | Only when n₁ > n₂; then θ₂ = 90° |
| θB | Brewster angle | Angle at which reflected light is fully polarised |
| v | Speed of light in medium | v = c / n |
Refractive Indices of Common Materials
| Material | n (≈) | Critical angle vs air | Brewster angle vs air |
|---|---|---|---|
| Vacuum | 1.0000 (by definition) | — | — |
| Air | 1.0003 | — | — |
| Water | 1.333 | 48.75° | 53.12° |
| Ethanol | 1.361 | 47.29° | 53.68° |
| Olive oil | 1.467 | 42.97° | 55.72° |
| Glycerin | 1.473 | 42.79° | 55.84° |
| Crown glass (BK7) | 1.517 | 41.24° | 56.60° |
| Polycarbonate | 1.585 | 39.13° | 57.75° |
| Flint glass (F2) | 1.620 | 38.12° | 58.30° |
| Sapphire | 1.770 | 34.42° | 60.53° |
| Diamond | 2.417 | 24.44° | 67.54° |
Frequently Asked Questions
The normal is perpendicular to the boundary and is the natural axis of symmetry for the refraction problem. Measuring angles from the surface itself would require trigonometry with the complementary angle (which turns sines into cosines) and would obscure the symmetry. Whenever you see Snell's law written with sines, the angles are always measured from the normal.
When light travels from a higher-index to a lower-index medium (n₁ > n₂), the refracted ray bends away from the normal. At some incidence angle θc = arcsin(n₂/n₁), the refracted ray grazes along the boundary at 90°. Beyond θc, no real refracted ray exists — the light is entirely reflected back into the first medium. This is total internal reflection. It is the basis of fibre-optic communication, prismatic binoculars, endoscopes, and the sparkle of a cut diamond.
At θ₁ = 0, Snell's law reduces to n₁ · 0 = n₂ · 0, which is true for every value of n₂. The two media are indistinguishable from the transmitted ray alone — it passes straight through regardless of the second index. To measure n₂ you must use a non-zero incidence angle, or measure it indirectly (for example by observing the critical angle from within the medium, or by using a prism refractometer).
Yes — the law is time-reversal symmetric. Reversing the direction of a ray traces the same path backwards, swapping which medium is "incident" and which is "refracting." The only asymmetry arises from total internal reflection: if a ray from dense to rare at θ₁ > θc is reflected, then a ray from rare to dense cannot produce TIR, because the corresponding ray would have to be incident at an angle whose sine exceeds 1.
Snell's law is the direct relationship between angles. Fermat's principle states that light travels between two points along the path that takes the least (or stationary) time. From Fermat's principle you can derive Snell's law — the derivation involves minimising the total travel time as a function of the crossing point on the boundary, and the resulting condition is exactly n₁ sinθ₁ = n₂ sinθ₂. Fermat's principle is the more general statement and also predicts reflection laws and curved paths in gradient-index media.
What it means
Snell’s law describes how light bends when passing from one medium to another with different refractive indices. It states that the product of the refractive index and the sine of the angle to the normal is constant across the interface. This law governs the behaviour of lenses, prisms, optical fibers, and the human eye. It is used in designing optical systems to control the path of light. Understanding Snell’s law is essential for opticians, physicists, and engineers working with any refractive optical system. It also explains natural phenomena like the apparent bending of a stick in water and mirages. The law is derived from Fermat’s principle of least time and is a cornerstone of geometrical optics.
Worked example
Snell's Law – Two Detailed Examples
Real‑World| Parameter | Value |
|---|---|
| n₁ | 1.00 |
| θ₁ (°) | 30 |
| n₂ | 1.50 |
| Parameter | Value |
|---|---|
| n₁ | 1.33 |
| θ₁ | 45 |
| n₂ | 1.00 |
Common mistakes
- Angle definition: θ₁ and θ₂ are measured from the normal to the interface – not from the surface.
- Indices of refraction n₁ and n₂: Must be ≥ 1 for real materials – n is dimensionless.
- Total internal reflection: Occurs when n₁ > n₂ and θ₁ exceeds the critical angle – the equation still holds but sinθ₂ > 1 (no real solution).
- Units: Angles in degrees or radians – ensure your calculator is in the correct mode.
- Wavelength dependence: n varies with wavelength (dispersion) – use the value for the specific wavelength.
Applications
Snell's law, n₁ sin θ₁ = n₂ sin θ₂, describes the relationship between the angles of incidence and refraction when light passes through an interface between two media with different refractive indices. This fundamental law of geometrical optics is the basis for understanding refraction, total internal reflection, and the design of lenses, prisms, and optical fibres. Engineers and physicists use it to calculate the bending of light in optical systems, to design corrective lenses, and to analyse the behaviour of light in fibres. It also underpins the working of refractometers and is used in oceanography to account for the apparent position of underwater objects. By applying Snell's law, professionals can optimise optical coatings, design imaging systems, and understand the dispersion of light in prisms. This law is essential for any work involving the transmission of light through different materials.
- Design of lenses, prisms, and optical instruments
- Calculation of refraction angles in optical systems
- Design of fibre optic cables for telecommunications
- Analysis of total internal reflection in waveguides
- Education on the physics of light and optics
Related calculators · Geometric Optics
Recently updated formulas
Suggest an improvement
Found a mistake? Have a better example? Let us know.
You may send up to 3 suggestions per 24 hours.