Snell's Law Refraction Calculator

Snell's law is n sin(θ) = n sin(θ), where n is refractive index and θ is measured from the normal. Plug in n, n, and θ; the calculator returns θ. So light passing from air (n = 1.00) into water (n = 1.33) at θ = 30° gives sin(θ) = sin(30°)/1.33 ≈ 0.376, so θ ≈ 22.1°. Light bends toward the normal entering a denser medium. Always measure angles from the normal (perpendicular to the surface), not from the surface itself — that's the most common student mistake.

Rearrange n sin(θ) = n sin(θ) to solve for either index. To find n from a known n and both angles: n = n sin(θ)/sin(θ). So light travelling from air at 45° refracts into an unknown medium at 30°: n = 1.00 × sin(45°)/sin(30°) = 0.707/0.5 ≈ 1.414. That's roughly water or a low-index glass. This is exactly how lab experiments measure refractive index — shine a laser, measure both angles, plug in. Calculators just speed up the trig. Reverse-solving is the basis for refractometers used in chemistry, food science, and gemmology.

When light goes from a denser medium to a less dense one (n > n), it bends away from the normal. At the critical angle θ_c, the refracted ray skims along the surface (θ = 90°). Setting sin(90°) = 1 gives sin(θ_c) = n/n. So at a glass-air interface (n = 1.5, n = 1.0): θ_c = arcsin(1/1.5) ≈ 41.8°. Above this angle, total internal reflection occurs — the basis of fibre optics and prismatic binoculars. The formula only works for n > n; otherwise no critical angle exists.

Provide both refractive indices and the angle of incidence θ; the tool returns θ via Snell's law. So light going from water (n = 1.33) into glass (n = 1.50) at 25° refracts to θ = arcsin(1.33 × sin(25°)/1.50) ≈ 22°. The bigger the index jump, the more the bend. Light always travels at a unique angle determined by the indices and the incident direction; there's no choice to be made. The calculator handles all the trig, including domain checks for total internal reflection cases where no real refraction angle exists.

Air has n ≈ 1.00 (technically 1.0003, close enough for most problems), and standard crown glass has n ≈ 1.50. So light entering glass at 45° refracts to sin(θ) = sin(45°)/1.50 ≈ 0.471, meaning θ ≈ 28.1°. Going the other way, light leaving glass at 28.1° refracts back to 45° — Snell's law is symmetric. Different glass types have different indices: flint glass around 1.62, fused silica around 1.46, dense flint over 1.7. Using the right index for the actual material matters for precision optics like camera lenses.

TIR happens when light tries to leave a denser medium at an angle greater than the critical angle θ_c = arcsin(n/n). At and beyond θ_c, no light refracts into the second medium — it all reflects back. Provide both indices and the incident angle, and the calculator tells you whether TIR occurs. Glass-air θ_c is about 41.8°, so a 45° incidence gives full reflection. This is the principle behind fibre-optic cables (light bounces along inside, never escaping) and the bright sparkle of diamonds (n ≈ 2.42, with θ_c ≈ 24.4°, so a lot of paths reflect totally inside).

Light slows down when entering a denser medium (higher refractive index). The wave's edge that enters first has its speed reduced before the rest catches up, and this differential slowing rotates the wavefront — like a marching band turning when one side hits soft ground. The result is the ray bending toward the normal. Higher index means slower light: glass slows light to about 2×10^8 m/s from 3×10^8 m/s. The reverse happens going from dense to thin medium — light speeds up and bends away from the normal. Snell's law captures this geometric consequence of changing wave speeds.