Snell's Law Calculator

Enter the refractive indices of two media and the angle of incidence to find the angle of refraction using Snell's Law (n1 sin θ1 = n2 sin θ2), plus the critical angle for total internal reflection.

Quick Facts

Snell's Law
n₁ sin θ₁ = n₂ sin θ₂
Relates the incidence and refraction angles to the refractive indices of the two media.
Critical angle
θc = arcsin(n₂ / n₁)
Only defined when n₁ > n₂ (light moving into a less dense medium); past θc, all light reflects.
Typical indices
Air ≈ 1.00, water ≈ 1.33, glass ≈ 1.52
A higher index means slower light speed in that medium and sharper bending toward the normal.

Your Results

Calculated
Angle of Refraction (θ₂)
-
From n₁ sin θ₁ = n₂ sin θ₂
sin(θ₂)
-
(n₁ / n₂) × sin θ₁
Critical Angle (θc)
-
Applies only when n₁ > n₂
Refraction Status
-
Transmitted ray or total internal reflection

Ready

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

About Snell's Law

Snell's Law (the law of refraction) describes how a light ray bends when it crosses the boundary between two transparent media with different optical densities. It states n₁ sin θ₁ = n₂ sin θ₂, where n₁ and n₂ are the refractive indices of the first and second media, θ₁ is the angle of incidence measured from the normal (the line perpendicular to the surface), and θ₂ is the angle of refraction. The relationship follows from Fermat's principle of least time and was formalized by Willebrord Snell in 1621.

Solving for the refraction angle

Rearranging Snell's Law gives θ₂ = arcsin[(n₁ / n₂) × sin θ₁]. Enter the refractive index of the medium the light starts in (n₁), the angle of incidence measured from the normal, and the refractive index of the medium it enters (n₂). The calculator computes sin θ₂ first, then converts it to the refraction angle in degrees. A higher refractive index means a denser optical medium, where light travels more slowly and bends more sharply toward the normal.

Total internal reflection and the critical angle

If (n₁ / n₂) × sin θ₁ works out to more than 1, no real angle θ₂ satisfies the equation — the light cannot refract into the second medium and instead reflects entirely back into the first. This can only happen when light travels from a denser medium into a less dense one (n₁ > n₂), and only beyond a threshold called the critical angle, θc = arcsin(n₂ / n₁). Total internal reflection is the principle behind fiber-optic cables, which guide light down a glass or plastic core by keeping every bounce beyond θc.

Common mistakes to avoid

  • Measuring from the surface instead of the normal: both θ₁ and θ₂ are always measured from the line perpendicular to the interface, not from the surface itself.
  • Swapping n₁ and n₂: n₁ belongs to the medium the light is leaving, and n₂ to the medium it is entering — reversing them inverts the direction of bending.
  • Forgetting total internal reflection: when n₁ > n₂, always check the incidence angle against the critical angle before assuming a refracted ray exists.

Frequently Asked Questions

What is Snell's Law?
Snell's Law relates how light bends when it crosses the boundary between two transparent media: n₁ sin θ₁ = n₂ sin θ₂, where n₁ and n₂ are the refractive indices of the two media and θ₁, θ₂ are the angles of incidence and refraction measured from the normal to the surface.
How do I find the critical angle for total internal reflection?
The critical angle is θc = arcsin(n₂ / n₁), and it only exists when light travels from a denser medium into a less dense one (n₁ > n₂). Past that angle of incidence, no light is transmitted — it all reflects back into the first medium.
What does it mean if sin θ₂ comes out greater than 1?
Since sine can never exceed 1, a computed value above 1 means there is no real solution for θ₂ — the light cannot refract into the second medium and undergoes total internal reflection instead.

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