Light changes direction when it crosses from one transparent material into another. That bending, called refraction, makes a straw look broken in a glass of water, lets lenses focus images and splits sunlight into rainbows. Snell’s law predicts exactly how much the ray bends from the refractive indices of the two materials. This calculator solves it for any unknown, flags total internal reflection, finds the critical angle and draws the incident, reflected and refracted rays.
How to use the Snell’s law calculator
- Choose what to Solve for: the angle of refraction, the angle of incidence, either refractive index, or the critical angle.
- Pick Medium 1, where the light comes from, and Medium 2, the one it enters. Presets cover vacuum, air, water, ice, ethanol, fused silica, acrylic, crown and flint glass, polycarbonate, sapphire and diamond. Choose “Custom index” to type your own value.
- Enter the angles measured from the normal, between 0° and 90°.
- Read the result alongside the deviation of the ray, the share of light reflected at the surface, Brewster’s angle, the speed of light in each medium and, when it exists, the critical angle.
Snell’s law formula
The refractive index is n = c ÷ v, the ratio of the speed of light in vacuum to its speed in the material. If n1 sin θ1 ÷ n2 comes out greater than 1, there is no refracted ray: the light is totally internally reflected. The fraction reflected at the surface comes from the Fresnel equations, averaged over both polarizations.
Worked example
Sunlight entering a pool
A ray in air (n = 1.000293) strikes still water (n = 1.333) at 45° from the normal.
sin θ2 = 1.000293 × sin 45° ÷ 1.333 = 0.53062
θ2 = arcsin(0.53062) = 32.05°
The ray bends 12.95° toward the normal. About 2.8% of the light reflects off the surface and the rest enters the water, where it travels at about 224,900 km/s.
From the other side. Reverse the media and look up at the surface from underwater at 60°. Now sin θ2 would be 1.154, which is impossible, so the calculator reports total internal reflection: the critical angle for water to air is 48.63°, and beyond it the surface acts as a perfect mirror. Divers see this as a silvery ceiling outside a circular “window” overhead.
Refractive indices of common materials
| Material | n (589 nm) | Critical angle into air |
|---|---|---|
| Water | 1.333 | 48.6° |
| Acrylic | 1.491 | 42.2° |
| Crown glass | 1.517 | 41.3° |
| Flint glass | 1.620 | 38.1° |
| Sapphire | 1.768 | 34.5° |
| Diamond | 2.417 | 24.4° |
Indices depend on wavelength, a property called dispersion. Values here are for yellow sodium light at 589 nm; blue light has a slightly higher index and bends more, which is why prisms and raindrops separate colors.
Practical uses
Optical fibers have a glass core surrounded by cladding of slightly lower index, so light hitting the boundary at a shallow angle is totally reflected and guided for kilometers. Refractometers measure sugar content in fruit juice or the salinity of seawater from the critical angle. Gemologists identify stones by their refractive index.
For lenses built on the same physics, use the thin lens and mirror equation calculator. To relate a light’s frequency and wavelength, or to see how the wavelength shortens inside a material, try the wavelength calculator. The inverse trig functions calculator explains the arcsin step.
Frequently asked questions
Are the angles measured from the surface or from the normal?
From the normal, the imaginary line perpendicular to the surface at the point where the ray hits. A ray that grazes the surface has an angle of incidence near 90°, and a ray that hits head-on has 0°. Measuring from the surface is the most common mistake with Snell's law.
What is the critical angle?
When light travels from a denser medium into a less dense one, there is an angle of incidence beyond which no light gets through: sin θc = n₂ ÷ n₁. For water into air it is 48.6°, for crown glass into air 41.3° and for diamond into air 24.4°.
What happens above the critical angle?
Total internal reflection: all the light reflects back into the first medium. Optical fibers trap light this way over long distances, and a diamond's low critical angle keeps light bouncing inside it, which makes it sparkle.
Why does light bend toward the normal when it enters water?
Light travels slower in water (about 225,000 km/s) than in air (about 299,700 km/s). The part of a wavefront that enters first slows down first, pivoting the wavefront toward the normal. Going the other way, it bends away.
What is Brewster's angle?
The angle of incidence, arctan(n₂ ÷ n₁), at which reflected light is completely polarized. Polarized sunglasses cut glare from water and roads because reflections near this angle are mostly horizontally polarized.