How to find the angle of refraction with Snell's law

How to · Updated 2026-11-10

Use n₁ sin i = n₂ sin r: multiply, divide by n₂, then take inverse sine. Worked example for air into water, plus how to spot total internal reflection.

To find the angle of refraction, rearrange Snell’s law to sin r = n₁ sin i / n₂ and take the inverse sine of the result; if that sine comes out bigger than 1, no refracted ray exists and the light is totally reflected. The snell's law page does this as you move the angle, draws the bent ray and prints the working line by line.

What you need

• The angle of incidence i, in degrees. It is measured from the normal, the imaginary line at right angles to the surface, not from the surface itself. If a question gives the angle to the surface, subtract it from 90° first.
• The refractive index of each material. n₁ is where the light starts, n₂ is where it goes. Typical values are air 1.00, water 1.33 and crown glass about 1.52. A bigger n means light travels more slowly in that material.
• A calculator in degree mode if you want to check by hand. Radian mode is the most common reason for a wildly wrong answer.

Step by step

1. Open the Snell’s law page and stay on Boundary. Two other views exist (Glass slab and Apparent depth), but the plain angle question lives here.
2. Pick the top and bottom media. The light travels from top to bottom. For an index that is not on the list, tick Type my own index and enter any value from 1 to 4.
3. Set the angle of incidence. Use the slider (0° to 89°, in whole degrees) or drag in the top half of the picture to aim the ray.
4. Read r. The large line gives the angle of refraction and says whether the ray bends towards or away from the normal. The box beneath shows the working, the deviation |i − r| and, when one exists, the critical angle.
5. Press Swap the two media to send the light the other way, for example from water out into air.

The formula behind it all:

n₁ sin i = n₂ sin r, so sin r = (n₁ × sin i) / n₂, and r = sin⁻¹(sin r)

One rule of thumb saves a lot of errors: going into a slower (higher n) material, r is smaller than i; going into a faster (lower n) material, r is larger.

A worked example

A torch beam in air hits a calm pond at 55° to the normal. What angle does it make with the normal under the water?

• n₁ = 1.00 (air), n₂ = 1.33 (water), i = 55°.
• sin 55° = 0.8192.
• sin r = 1.00 × 0.8192 / 1.33 = 0.6159.
• r = sin⁻¹(0.6159) = 38.0°.

The beam bends 17.0° towards the normal, as it should when going into the slower material. Set air on top, water below and the slider to 55°, and the page reads r = 38° (it rounds to three significant figures, so the trailing zero is dropped).

Now turn it round. A light at the bottom of the pond shines up at the surface at 50° to the normal. Press Swap the two media:

• sin r = 1.33 × sin 50° / 1.00 = 1.33 × 0.7660 = 1.019.

No angle has a sine above 1, so there is no refracted ray at all. All of the light reflects back down: total internal reflection. The page says so and gives the critical angle for water into air, C = sin⁻¹(1/1.33) = 48.8°. Any underwater ray steeper than that stays trapped. Move the slider below 48.8° and the ray escapes again, bent away from the normal.

Want to see the same law split white light into colours? The prism and dispersion page applies it twice, once at each face of a prism. For curved glass surfaces, the lens ray diagram takes over.

Check your answer

• Put it back in. 1.33 × sin 38.0° = 0.8188, which matches sin 55° = 0.8192 to three figures. Both sides of Snell’s law should agree.
• Direction. Into water, r must be smaller than i. 38.0° is less than 55°, so the bend is the right way.
• Straight on. At i = 0°, r is 0° whatever the materials. If your method gives anything else at 0°, the setup is wrong.
• Too-big sines. A sine above 1 is not a mistake to round away. It is the sign of total internal reflection, and it can only happen going from a higher n to a lower n.

Questions people ask

Do I measure the angle from the surface or the normal? From the normal, every time. An angle of 30° to the surface is 60° to the normal.

Does the colour of the light matter? Slightly. Glass and water bend violet a little more than red, because n changes with wavelength. For school questions one rounded value of n is used.

How do I find the refractive index from two angles? Rearrange the same law: n₂ = n₁ sin i / sin r. With air as the first medium, that is simply sin i / sin r.