Angle of Refraction Calculator

Apply Snell's Law (n₁ sin θ₁ = n₂ sin θ₂) to find how far a light ray bends when it crosses between two media with different refractive indices.

Quick Facts

Formula
Snell's Law: n₁ · sin θ₁ = n₂ · sin θ₂
Light bends toward the normal entering a denser medium (n₂ > n₁) and away from it entering a less dense one; if n₁ > n₂ and θ₁ exceeds the critical angle, total internal reflection occurs instead of refraction.

Your Results

Calculated
Angle of refraction (θ₂)
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Angle from the normal inside medium 2
Deviation
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How far the ray bends: |θ₁ − θ₂|
Critical angle (θc)
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Beyond this angle, total internal reflection occurs (only when n₁ > n₂)
Relative index (n₂ / n₁)
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Equals sin θ₁ / sin θ₂ by Snell's Law

Ready

Enter the incidence angle and both refractive indices, then press Calculate.

Understanding the Angle of Refraction

When a beam of light crosses the boundary between two transparent media — air to water, air to glass, glass to air — it bends. The angle of refraction is the angle between the refracted ray and the normal (an imaginary line perpendicular to the surface) inside the second medium. This calculator finds that angle from the angle of incidence and the refractive indices of the two media using Snell's Law.

The formula: Snell's Law

Snell's Law relates the two angles to the refractive indices of the media:

n₁ · sin θ₁ = n₂ · sin θ₂

where n₁ is the refractive index of the medium the light starts in, θ₁ is the angle of incidence, n₂ is the refractive index of the medium the light enters, and θ₂ is the angle of refraction — all angles measured from the normal, not the surface. Solving for the unknown angle gives:

θ₂ = arcsin[(n₁ / n₂) · sin θ₁]

Which way does light bend?

  • Entering a denser medium (n₂ > n₁): light slows down and bends toward the normal, so θ₂ < θ₁ — for example, light entering water from air.
  • Entering a less dense medium (n₂ < n₁): light speeds up and bends away from the normal, so θ₂ > θ₁ — for example, light exiting glass into air.
  • Equal indices (n₂ = n₁): the ray passes straight through with no bending at all.

Total internal reflection and the critical angle

When light travels from a denser medium into a less dense one (n₁ > n₂), there is a limit. Past a certain incidence angle, called the critical angle θc = arcsin(n₂/n₁), Snell's Law has no solution for θ₂ — the light cannot refract out at all and instead reflects entirely back into the first medium. This is total internal reflection, the effect that keeps light trapped inside fiber-optic cables and gives cut diamonds their sparkle.

Typical refractive indices

  • Vacuum: 1.0000 (by definition)
  • Air (at sea level): about 1.0003, commonly rounded to 1.00
  • Water: about 1.33
  • Crown glass: about 1.52
  • Diamond: about 2.42

Frequently Asked Questions

What is the formula for the angle of refraction?
Snell's Law states n1 sin θ1 = n2 sin θ2, where n1 and θ1 describe the incident medium and ray, and n2 and θ2 describe the refracting medium and ray. Solving for the refraction angle gives θ2 = arcsin[(n1/n2) sin θ1], with all angles measured from the normal to the surface, not from the surface itself.
Why does light bend when it enters a new medium?
Light bends because its speed changes when it crosses into a medium with a different refractive index (speed = c / n). That change in speed bends the wavefront toward the normal when the light slows down entering a denser medium, and away from the normal when it speeds up entering a less dense one.
What is total internal reflection and when does it happen?
Total internal reflection happens when light traveling in a denser medium (n1 greater than n2) hits the boundary at an angle beyond the critical angle, θc = arcsin(n2/n1). Past that angle, Snell's Law has no valid solution for the refraction angle, so all the light reflects back into the first medium instead of exiting.
Does the angle of refraction depend on the color of the light?
Yes, slightly. A material's refractive index varies a little with wavelength, an effect called dispersion, so red and blue light refract at marginally different angles through the same material. This calculator uses a single refractive index per medium; for precise multi-wavelength work, look up the index at the specific wavelength you need.

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