Refraction calculator (Snell's law)

A Snell's law calculator works out how much light bends when it passes from one material into another: n₁ sin θ₁ = n₂ sin θ₂, where n is the refractive index and θ the angle from the normal. Light entering glass from air at 40° bends to 25.0°. Our Refraction Calculator finds the angle of refraction or incidence, a refractive index, the speed of light in a material, or the critical angle, with the working and a ray diagram.

It opens on Snell's law: light going from air into crown glass at 40°.

Try it

Refraction Calculator

Use Snell's law to find angles, refractive index and the critical angle

Open the calculatorGo to its page

How to use the Snell's law calculator

1. Under Calculate, choose a group: Snell's law, Index or Critical.
2. In the second row, choose what to find. Snell's law has Refracted, Incident and Index n₂; Index has Index (from speed) and Speed; Critical has the critical angle and the index from it.
3. Type the values, or pick a material from Common values: air, ice, water, ethanol, Perspex, crown glass, flint glass, sapphire or diamond.
4. Read the answer, the working and the extra readings: how far the ray bends and which way, and the speed of light in each material.

If the angle is beyond the critical angle, the calculator says total internal reflection happens instead, and gives the critical angle. Practice and help sets refraction questions with worked solutions. Our Snell's law guide covers the law in depth.

The refraction formulas

Angles are measured from the normal, the line at right angles to the surface. Going into a denser material (higher n), light slows down and bends towards the normal; coming out, it bends away.

Worked examples you can check in the calculator

Example 1: air into glass

The calculator gives 25.02°: the ray bends 14.98° towards the normal, and light slows from 2.997 × 10⁸ m/s to 1.97 × 10⁸ m/s.

Example 2: water into air

Light leaves water (n = 1.33) at 30° to the normal. Choose water for n₁ and air for n₂: the calculator gives 41.67°, bending away from the normal, and notes the critical angle, 48.77°.

Example 3: total internal reflection

Make the angle in water 60°. Now sin θ₂ would be 1.33 × sin 60° ÷ 1.0003 = 1.15, more than 1, so no ray can cross. The calculator says total internal reflection: all the light reflects at 60°, above the critical angle of 48.77°.

Example 4: finding a refractive index

A ray enters a block from air at 45° and refracts to 28°. Choose Index n₂:

The calculator gives 1.5066, which suggests a glass or Perspex block, and says light travels at 1.99 × 10⁸ m/s inside it.

Try Snell's law in the course, then press Mark:

Example 5: diamond

Diamond has n = 2.42, so light travels at 2.998 × 10⁸ ÷ 2.42 = 1.24 × 10⁸ m/s in it. Its critical angle with air is 24.4°, very small, so light inside a cut diamond reflects many times before it leaves. That is part of why diamonds sparkle.

What the calculator does not do

• One boundary at a time. For light through a glass block or a prism, apply it at each surface in turn, or use the Ray Optics Simulator.
• No colour dependence. It uses one index per material, so it does not show dispersion into a spectrum.
• No lenses. For focal length and images, use the lens equation.

Common mistakes

• Measuring from the surface. Angles are from the normal, not the surface.
• Swapping n₁ and n₂. n₁ belongs to the material the light starts in.
• Using the critical angle formula the wrong way up. sin θc = n₂ ÷ n₁ with the denser material on the bottom, so the answer is less than 1.
• Calculator in radians. Use degree mode for sin and sin⁻¹.

Practise light in a short timed paper

Refraction and the critical angle come up in A-level Physics: Year 1 and GCSE Physics Higher practice. Each daily paper is a short timed paper of 8 questions worth 25 to 35 marks, and the papers mix topics. Our total internal reflection guide and critical angle formula guide go further.

Read next

Physics · Waves and lightSnell's law: n₁ sin θ₁ = n₂ sin θ₂ explained, with examplesSnell's law n₁ sin θ₁ = n₂ sin θ₂ explained: refractive index, n = sin i ÷ sin r, bending towards or away from the normal, worked examples and practice.