Snells Law Calculator
The bend at every boundary: n₁·sinθ₁ = n₂·sinθ₂ refracts your ray, names the critical angle, and calls total internal reflection by name instead of crashing.
Snells Law Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- Isotropic media, single wavelength
- No partial reflectance fractions
In short: Light leaves air (n = 1.00) and hits glass (n = 1.50) at 30° from the normal: sinθ₂ = 0.333333, so θ₂ = 19.471221° — bent TOWARD the normal, paying the speed tax of the denser medium (c/1.5 ≈ 199,861,639 m/s against 299,792,458 in vacuum). There is no critical angle entering a denser medium; leaving the glass it is 41.810315° — and past that, total internal reflection, the trick that keeps fibre light and diamonds sparkly.
Formula
n₁·sinθ₁ = n₂·sinθ₂ · θ₂ = arcsin(n₁ sinθ₁ / n₂) · θc = arcsin(n₂/n₁), n₁ > n₂ only
Angles are measured from the NORMAL, never the surface — the classic slip. The index n is the slowdown factor: light crawls at c/n in a medium, and the bending is exactly the cost of the speed change at an angle. Entering a denser medium the ray bends toward the normal; leaving, away. When the leaving ray would need a sine bigger than 1, no refraction exists and the boundary reflects EVERYTHING — total internal reflection, a physical answer this page prints rather than an error it hides.
Worked Example
- Enter both indices (vacuum is exactly 1, the floor).
- Enter the incidence angle from the normal.
- Read the refracted angle and which way it bent.
- Read the critical angle — or the TIR card when your ray is past it.
Defaults: 30° air to glass → 19.471221° toward the normal; critical angle out of glass 41.810315°. At 45° glass to air: total internal reflection.
Strengths & Limits Of This Model
Where this engine is strong
- TIR named and carded, never faked
- Critical angle computed both directions
Where it stops
- No dispersion splitting
- No Fresnel reflectance percentages
Practical Use Cases
Fibre optics
why the light stays in the glass
Gem work
the critical angle that traps sparkle
Teaching
refraction without the folklore
Methodology & Editorial Standards
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Snells Law Calculator — 8 Expert FAQs
8 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
Why measure from the normal and not the surface?
Because the law is written in sines of angles FROM the normal — the perpendicular to the surface. Measuring from the surface sneaks a 90° minus into every term and flatters the wrong angle. Every angle on this page, in the diagram tradition and in your optics textbook, leans on that one perpendicular.
What is the critical angle, physically?
The escape threshold. Going from dense to thin, refracted rays bend AWAY from the normal; at the critical angle the refracted ray skims the surface at exactly 90°. Past it there is nowhere for the ray to go — the boundary returns everything. Only dense-to-thin transitions have one: air to glass has none.
Why does total internal reflection reflect EVERYTHING?
Because there is no transmitted angle that satisfies the law: n₁·sinθ₁ / n₂ exceeds 1 and arcsin refuses. Reflection takes the remainder — all of it. This is engineering's cleanest mirror: no metallic coating, no losses beyond the glass itself, which is why fibre optics and prism binoculars run on it.
Why do diamonds sparkle more than glass?
Their index is higher (≈ 2.417 against 1.5), so the critical angle is smaller (≈ 24.4° against 41.8°) — a cut facet traps light through a wider range of strikes and bounces it around more before releasing it toward your eye. The sparkle is geometry on this page's arithmetic, cut into the stone.
Does Snell's law apply to sound or only light?
To waves in general — any wave changing speed across a boundary at an angle bends the same way, with an effective index from the speeds. Sound refracts in the ocean and the atmosphere; the wave-speed page supplies the pair of speeds and the same sine bookkeeping finishes the job.
What is n, really?
The ratio of light's speed in vacuum to its speed in the medium — vacuum is exactly 1 by definition, air about 1.00, water about 1.333, glasses about 1.5, diamond about 2.417. All are approximate material constants; they drift with temperature and colour (dispersion — which is why prisms split white light).
What happens exactly at the normal incidence, θ₁ = 0?
Nothing to bend: sin 0 = 0, so θ₂ = 0 and the ray goes straight through, direction unchanged. The speed STILL changes — the wavelength absorbs the difference (the wavelength page's card) — but the direction does not. Perpendicular arrival is the honest zero of this law.
Is light the only thing that has a critical angle?
No — any wave leaving a slower medium for a faster one at a shallow angle can be totally reflected. Ocean acoustics channels whale song with it; earthquake waves bounce off the Moho with it. Swap the indices for speed ratios and this page's arithmetic is general wave machinery.