About this calculator
When light crosses from one material into another it changes speed and bends. Snell's law says by how much: n₁ sin θ₁ = n₂ sin θ₂.
Enter the refractive index of each material and the angle of the incoming ray, measured from the normal (an imaginary line at right angles to the surface). You also get the critical angle and the speed of light in each material.
Worked examples
Real numbers, worked out by the same calculator. Press “Use these numbers” to try one above.
Air into glass at 30°
- Angle of refraction
- 19.471221°
- Critical angle
- None: light is entering a denser (or equal) material
- Speed of light in medium 1
- 299,792,458 m/s
- Speed of light in medium 2
- 199,861,638.667 m/s
Light hitting the boundary at 30° bends to 19.471221° (towards the normal).
Show the working
- Snell's law: n₁ sin θ₁ = n₂ sin θ₂, so sin θ₂ = 1 × sin(30°) ÷ 1.5 = 0.333333.
- θ₂ = arcsin(0.333333) = 19.471221°.
- Light entering a denser material always bends towards the normal, so there is no critical angle.
Glass into air at 60° (total reflection)
- Angle of refraction
- None: total internal reflection
- Critical angle
- 41.810315°
- Speed of light in medium 1
- 199,861,638.667 m/s
- Speed of light in medium 2
- 299,792,458 m/s
At 60° the light is totally internally reflected, because it is beyond the critical angle of 41.810315°.
Show the working
- Snell's law: n₁ sin θ₁ = n₂ sin θ₂, so sin θ₂ = 1.5 × sin(60°) ÷ 1 = 1.299038.
- That is more than 1, which is impossible, so the light cannot pass through: it is totally reflected.
- Critical angle = arcsin(n₂ ÷ n₁) = arcsin(1 ÷ 1.5) = 41.810315°. Beyond it, all the light is reflected.
Air into water at 45°
- Angle of refraction
- 32.117631°
- Critical angle
- None: light is entering a denser (or equal) material
- Speed of light in medium 1
- 299,792,458 m/s
- Speed of light in medium 2
- 225,407,863.158 m/s
Light hitting the boundary at 45° bends to 32.117631° (towards the normal).
Show the working
- Snell's law: n₁ sin θ₁ = n₂ sin θ₂, so sin θ₂ = 1 × sin(45°) ÷ 1.33 = 0.531659.
- θ₂ = arcsin(0.531659) = 32.117631°.
- Light entering a denser material always bends towards the normal, so there is no critical angle.
Water into air at 40°
- Angle of refraction
- 58.74952°
- Critical angle
- 48.753467°
- Speed of light in medium 1
- 225,407,863.158 m/s
- Speed of light in medium 2
- 299,792,458 m/s
Light hitting the boundary at 40° bends to 58.74952° (away from the normal).
Show the working
- Snell's law: n₁ sin θ₁ = n₂ sin θ₂, so sin θ₂ = 1.33 × sin(40°) ÷ 1 = 0.854908.
- θ₂ = arcsin(0.854908) = 58.74952°.
- Critical angle = arcsin(n₂ ÷ n₁) = arcsin(1 ÷ 1.33) = 48.753467°. Beyond it, all the light is reflected.
Typical refractive indices
- Vacuum: 1 (air is about 1.0003)
- Water: about 1.33
- Glass: about 1.5 (varies from 1.45 to 1.9)
- Diamond: about 2.42
The index is the speed of light in a vacuum divided by its speed in the material, so light travels at c ÷ n.
Total internal reflection
Going from a denser material to a less dense one, the ray bends away from the normal. Beyond the critical angle, θc = arcsin(n₂ ÷ n₁), it cannot escape at all and is reflected completely. This is how optical fibres trap light and why diamonds sparkle.
Measuring from the normal
Angles in Snell's law are measured from the normal, not from the surface. A ray hitting a surface at 30° to the surface has an angle of incidence of 60°. A ray along the normal (0°) passes straight through without bending.
Frequently asked questions
What is Snell's law?
n₁ sin θ₁ = n₂ sin θ₂: the relationship between the angles and the refractive indices on either side of a boundary.
What is the normal?
A line at right angles to the surface. All angles are measured from it.
What is the critical angle?
The incidence angle above which light going into a less dense material is totally reflected.
Why does the light bend?
Because it changes speed. Slower in the denser material, so the ray turns towards the normal.
What if the answer says none?
Then light cannot get through at that angle: it is totally internally reflected.
Formulas tested against hand-worked answers. Last reviewed 29 September 2026. These calculators do arithmetic only; they are not financial, tax or legal advice.