This Snell’s law calculator finds how light bends crossing between materials — the refraction angle from the incident angle and both refractive indices — and flags the critical angle where bending becomes total internal reflection.
The Law and the Speed Behind It
n₁ sin θ₁ = n₂ sin θ₂, with angles measured from the normal (perpendicular to the surface — the classic gotcha). The index n is a speed ratio: light travels at c/n in the medium, and the bend is nature keeping the wavefronts connected across the boundary, like a marching band wheeling when one side steps onto mud. Entering a denser medium (higher n): slows, bends toward the normal. Leaving for a rarer medium: speeds up, bends away — until, past the critical angle, it cannot leave at all and reflects internally at 100% efficiency that no mirror matches. The bending inside lenses is engineered on our Lens Calculator. Part of our Physics Calculators collection.
Worked Examples (Verified)
- Air → water at 45°: sin θ₂ = sin 45° ÷ 1.333 = 0.5305 → θ₂ = 32.0° — why a pool floor looks shallower: your brain unbends the ray that bent
- Water → air critical angle: asin(1 ÷ 1.333) = 48.6° — a diver looking up past 48.6° from vertical sees the surface become a mirror reflecting the pool (Snell’s window is the 97° cone inside it)
- Air → diamond at 45°: θ₂ = asin(0.7071 ÷ 2.42) = 17.0° — and diamond’s critical angle is just 24.4°, so cut stones trap light through multiple internal reflections: the sparkle is Snell’s law with good tailoring
Common Refractive Indices
| Material | n | Material | n |
|---|---|---|---|
| Vacuum / air | 1.000 | Crown glass | 1.52 |
| Water | 1.333 | Flint glass | 1.62 |
| Ethanol | 1.36 | Sapphire | 1.77 |
| Vegetable oil | 1.47 | Diamond | 2.42 |
Indices vary slightly with wavelength (dispersion — blue bends more than red), which is both the rainbow and the chromatic aberration lens designers fight.
Related Tools
- Lens Calculator — refraction put to work
- Wavelength Calculator — the wave being bent
- Physics Calculators — the full collection
Frequently Asked Questions
How do you use Snell’s law?
n₁ sin θ₁ = n₂ sin θ₂, angles from the normal. Solve for sin θ₂ = (n₁/n₂) sin θ₁ and take the arcsine — air to water at 45° gives 32.0°. This calculator does it with a critical-angle check built in.
What is the critical angle?
The incident angle (going dense → rare) at which the refracted ray would graze 90°: θc = asin(n₂/n₁). Beyond it, refraction is impossible and total internal reflection occurs — 48.6° for water→air, 24.4° for diamond→air.
Why does a straw look bent in a glass of water?
Light from the submerged part refracts away from the normal leaving water (32°-style geometry in reverse), and your visual system, assuming straight rays, reconstructs the straw shallower and offset. Every apparent-depth effect is Snell’s law plus a brain that refuses to model it.
How do fiber optics work?
A core of high-n glass clad in lower-n glass: light aimed steeply enough strikes every internal boundary beyond the critical angle and reflects perfectly, zig-zagging kilometers with barely any loss. Total internal reflection as infrastructure.
Does the frequency of light change when it refracts?
No — frequency is fixed by the source; speed and wavelength change together in the medium (λ_medium = λ_vacuum ÷ n). The color you perceive stays the same, which is why a red laser is red underwater.
Is this Snell’s law calculator free?
Yes — free, instant, no sign-up.