Snell's Law Refraction Calculator
Snell's law describes how a light ray bends when crossing the boundary between two media: n₁ sin θ₁ = n₂ sin θ₂. Pick the two refractive indices, enter the incidence angle, and the tool returns the refraction angle. When light goes from a denser to a less dense medium (n₁ > n₂), it also shows the critical angle and flags total internal reflection.
Medium 1 (incident side)
Medium 2 (refraction side)
Enter positive refractive indices and an incidence angle between 0° and 90°.
Refraction angle θ₂
22.03°
θ₂
Critical angle θ_c
—
Going into a denser medium (n₁ ≤ n₂) — there is no critical angle; the ray always refracts.
Phase-velocity ratio v₂ / v₁
0.7502
From v = c / n. A value below 1 means light is slower in medium 2 than in medium 1.
Working
n₁ sin θ₁ = n₂ sin θ₂
Default is 30° from air into water — refraction angle ~22°. Change the media or angle to see the critical angle and TIR notice update.
Formula
n₁ sin θ₁ = n₂ sin θ₂ → θ₂ = arcsin((n₁/n₂) sin θ₁) ; critical angle θ_c = arcsin(n₂/n₁) (only when n₁ > n₂)
- · All angles are measured from the normal (the line perpendicular to the surface), not from the surface itself.
- · Refractive index is defined as n = c / v: speed of light in vacuum divided by phase velocity in the medium. Ordinary transparent materials all have n ≥ 1.
- · When n₁ > n₂ (dense → less dense) and θ₁ > θ_c, the ray cannot refract out and is wholly reflected back — that's total internal reflection (TIR), the mechanism that traps light inside an optical fibre.
- · When n₁ ≤ n₂ (less dense → dense) there is no critical angle; the ray always refracts.
- · Default indices are tabulated at the sodium-D wavelength (λ ≈ 589 nm) at 20 °C. Sources: CRC Handbook of Chemistry and Physics; Hecht, "Optics" (Table 4.1). Real values vary slightly with wavelength (dispersion) and temperature.
- · The tool assumes a flat interface, monochromatic light and no polarisation effects — fine for standard high-school and first-year university optics. Birefringent crystals (e.g. calcite) and non-linear media need more advanced treatment.
Frequently asked
Why do my legs look shorter or bent when I look into water?
Light from your foot refracts as it leaves the water: n_water (≈1.333) sin θ_water = n_air (≈1.0) sin θ_air. Because n_water > n_air, θ_air is larger than θ_water and the ray bends away from the normal. The virtual image you see is pulled closer to the surface and appears foreshortened. Try the tool with medium 1 = water, medium 2 = air, θ₁ = 30° — you should see a refraction angle close to 42°.
Why does the critical angle and total internal reflection matter in real life?
Total internal reflection is what keeps light inside an optical fibre over tens of kilometres — light hits the core/cladding boundary above the critical angle and reflects perfectly every time. Periscopes and binoculars use right-angle prisms instead of mirrors for the same reason: the glass-to-air critical angle is about 41°, so a 45° ray reflects with essentially 100% efficiency — brighter and more durable than any silvered mirror. The 24.4° critical angle of diamond is also why cut diamonds sparkle — light bounces around inside several facets before finally escaping.
Are the incidence and refraction angles measured from the surface itself?
No — both angles are measured from the normal, the line perpendicular to the surface. If a question gives the grazing angle (measured from the surface itself), subtract from 90° before plugging in. "Light hits the surface at 60° from the surface" means an incidence angle of 30° from the normal.
Why do different colours bend by different amounts? (dispersion)
The refractive index itself depends on wavelength — a phenomenon called dispersion. Blue light (shorter wavelength) has a slightly higher index than red, so blue bends more in a prism or a raindrop. That is what splits sunlight into a rainbow and a prism into its spectrum. The tool uses a single value at the sodium-D wavelength; for advanced work, look up the index at your specific wavelength and plug it in.
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