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Snell's Law Calculator
Solve n₁ sin θ₁ = n₂ sin θ₂ for any unknown: the refraction angle, the incidence angle or either refractive index. Pick materials like air, water, glass or diamond, see a ray diagram, and get the critical angle, a total internal reflection check, the Brewster angle and how much light is reflected.
Last updated:
Solve for
°
Measured from the normal (perpendicular), 0–90°
°
Enter your values to see the result.
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How to use the Snell's Law Calculator
- Choose what to solve for.
- Pick the two media or enter custom refractive indices.
- Enter the known angle(s), measured from the normal.
- Read the result, check the ray diagram and the critical angle.
Snell's law formulas
| Find | Formula |
|---|---|
| Refraction angle | θ₂ = arcsin(n₁ sin θ₁ / n₂) |
| Incidence angle | θ₁ = arcsin(n₂ sin θ₂ / n₁) |
| Refractive index | n₂ = n₁ sin θ₁ / sin θ₂ |
| Critical angle (n₁ > n₂) | θc = arcsin(n₂ / n₁) |
| Brewster angle | θB = arctan(n₂ / n₁) |
| Speed of light in a medium | v = c / n |
Example: light going from air (n = 1.000293) into water (n = 1.333) at 45°: sin θ₂ = 1.000293 × 0.7071 / 1.333 = 0.5306, so θ₂ = 32.05°.
Refractive indices of common materials (≈ 589 nm)
| Material | n | Critical angle to air |
|---|---|---|
| Air | 1.000293 | — |
| Ice | 1.309 | 49.8° |
| Water | 1.333 | 48.6° |
| Ethanol | 1.361 | 47.3° |
| Fused quartz | 1.458 | 43.3° |
| Acrylic (PMMA) | 1.49 | 42.2° |
| Crown glass | 1.52 | 41.2° |
| Polycarbonate | 1.586 | 39.1° |
| Flint glass | 1.62 | 38.1° |
| Sapphire | 1.77 | 34.4° |
| Cubic zirconia | 2.15 | 27.7° |
| Diamond | 2.417 | 24.4° |
Frequently asked questions
What is Snell's law?
Snell's law describes how light bends at a boundary: n₁ sin θ₁ = n₂ sin θ₂, where n₁ and n₂ are the refractive indices and θ₁, θ₂ the angles measured from the normal (the line perpendicular to the surface).
What is the critical angle?
When light goes from a higher to a lower refractive index, the critical angle θc = arcsin(n₂ / n₁) is the incidence angle at which the refracted ray grazes the surface (θ₂ = 90°). For water to air it is about 48.6°, for crown glass to air about 41.2° and for diamond to air about 24.4°.
What is total internal reflection?
If the angle of incidence exceeds the critical angle, Snell's law has no solution (sin θ₂ would be greater than 1) and all the light reflects back into the denser medium. Optical fibres and the sparkle of diamonds rely on it.
Does light bend toward or away from the normal?
Toward the normal when it enters a medium with a higher refractive index (air into water) and away from the normal when it enters a lower one (water into air).
What is the Brewster angle?
θB = arctan(n₂ / n₁). At this angle the reflected light is completely polarised parallel to the surface (p-polarised reflection is zero). For air to glass (n = 1.52) it is about 56.7°.
Why do the refractive indices differ slightly between sources?
The index depends on wavelength and temperature. The presets use typical values for yellow light (about 589 nm) at room temperature; enter your own value for other conditions.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.
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