Refraction angles, index of refraction from measured angles, the critical angle and total internal reflection, checked against OpenStax University Physics Volume 3
How Snell’s Law Works
Light changes speed when it crosses into another material, v = c/n, and that change bends its path. The index of refraction n is the ratio of the speed of light in a vacuum to its speed in the material. Snell’s law, n₁ sin θ₁ = n₂ sin θ₂, relates the angles on each side, measured from the normal (the line perpendicular to the surface).
- Multiply n₁ by the sine of the incident angle: 1.00 × sin 30.0° = 0.500.
- Divide by n₂: 0.500 ÷ 2.419 = 0.207.
- Take the inverse sine: θ₂ = 11.9°, as in OpenStax Example 1.3.
Run it backward to find an unknown index. In Example 1.2, light from air at 30.0° refracts to 22.0°: n₂ = 1.00 × sin 30.0° ÷ sin 22.0° = 1.33, the index of water.
Refraction Angles at Common Interfaces
Going from water into air, the light bends away from the normal, and above about 48.8° it can’t get out at all.
Critical Angle and Total Internal Reflection
When light goes from a higher index to a lower one, the refraction angle reaches 90° at the critical angle θc = sin⁻¹(n₂/n₁). Beyond it, all the light reflects back inside. OpenStax Example 1.4 finds 42.2° for polystyrene in air: sin⁻¹(1.00/1.49).
Diamond’s small critical angle traps light inside a well-cut stone, which is why it sparkles. Total internal reflection is also what keeps light inside optical fibers.
Common Mistakes
- Measuring angles from the surface. Snell’s law uses angles from the normal.
- Calculator in radians. Make sure sine and inverse sine use degrees.
- Looking for a critical angle going into a denser medium. Total internal reflection needs n₁ > n₂; from air into water it can’t happen.
- Flipping n₂/n₁. The critical angle is sin⁻¹(n₂/n₁) with n₂ the lower index, so the ratio is below 1.
What This Calculator Does Not Cover
It treats n as a single number. Real indices change slightly with wavelength (dispersion, which makes rainbows), and the calculator does not compute how much light is reflected versus transmitted at angles below the critical angle.
Frequently Asked Questions
What is Snell’s law?
n₁ sin θ₁ = n₂ sin θ₂: the product of the index of refraction and the sine of the angle from the normal is the same on both sides of a boundary.
How do you calculate the critical angle?
θc = sin⁻¹(n₂/n₁), where light travels from the higher index n₁ toward the lower index n₂. For polystyrene to air it is sin⁻¹(1.00/1.49) = 42.2°.
When does total internal reflection happen?
When light goes from a higher to a lower index of refraction and the incident angle is larger than the critical angle.
Does light bend toward or away from the normal?
Toward the normal when entering a higher index (slower) material, away from it when entering a lower index one.
What is the index of refraction of water?
About 1.33, as OpenStax Example 1.2 finds from a 30.0° incident angle and a 22.0° refraction angle.
Why does a diamond sparkle?
Its high index (2.419) gives a critical angle of only about 24°, so light entering the stone reflects internally many times before leaving.
How to Use This Calculator
- Choose what to find — the refraction angle, or the second index from two measured angles.
- Enter the indices — n₁ for the medium the light comes from, n₂ for the one it enters.
- Enter the angle of incidence — in degrees from the normal (and the measured refraction angle in index mode).
- Read the results — the refraction angle or total internal reflection, the critical angle and the speed of light in each medium.
📚 Sources & References- [1] OpenStax. University Physics Volume 3, 1.3 Refraction (2016; accessed October 2026). openstax.org
- [2] OpenStax. University Physics Volume 3, 1.4 Total Internal Reflection (2016; accessed October 2026). openstax.org
✅
Editorial Standards — Every calculator is built from peer-reviewed formulas and official data sources, editorially reviewed for accuracy, and updated regularly.
Read our full methodology ·
About the author