Last updated: 2026-06-14

Snell's Law

Snell's law describes how a ray of light changes direction when it crosses the boundary between two transparent materials. It relates the angle of incidence to the angle of refraction through the refractive indices of the two media: $n_1 \sin\theta_1 = n_2 \sin\theta_2$. The angles are measured from the normal — the line drawn perpendicular to the surface at the point where the ray hits it. The relationship was described by Willebrord Snellius in 1621 and is the foundation of geometric optics, governing lenses, prisms, fibre optics and the apparent bending of a straw in a glass of water.

This calculator returns the angle of refraction from the two indices and the incidence angle, and flags total internal reflection when it occurs.

Bending toward or away from the normal

Whether the ray bends toward or away from the normal depends only on which medium is optically denser. When light enters a denser medium — one with a higher refractive index, such as going from air into glass — it slows down and bends toward the normal, so the refraction angle is smaller than the incidence angle. Going the other way, from glass into air, it speeds up and bends away from the normal. If both media have the same index there is no bending at all.

Formula

The refractive index of a material is the ratio of the speed of light in vacuum to its speed in the material. Vacuum is exactly 1, air is 1.0003, water is 1.33, ordinary glass is about 1.5 and diamond is 2.42.

Worked example

A ray of light in air ($n_1 = 1.00$) strikes a flat sheet of glass ($n_2 = 1.50$) at an angle of incidence of 30°. The refraction angle is:

The ray bends toward the normal because glass is denser than air. Entering 1.00, 30° and 1.50 reproduces this result.

Total internal reflection

When light travels from a denser medium into a less dense one ($n_1 > n_2$), there is a limit to how large the incidence angle can be before refraction becomes impossible. At the critical angle $\theta_c = \arcsin(n_2/n_1)$, the refracted ray would graze along the surface at 90°. Beyond that angle, Snell's law would require $\sin\theta_2$ to exceed 1, which cannot happen, so all the light reflects back into the first medium. This total internal reflection is what traps light inside an optical fibre and what makes a diamond sparkle. For a water-to-air boundary the critical angle is about 48.6°; for glass to air it is about 41.8°.

Limitations

Snell's law applies to the boundary between two uniform, transparent media and treats light as a ray. It does not by itself tell you how much light is reflected versus transmitted — that requires the Fresnel equations — and it ignores the slight dependence of the refractive index on wavelength, which is what splits white light into a spectrum in a prism. For that dispersion effect, each colour must be traced with its own index.