Is the angle measured from the edge or normal?

To measure the normal, construct a line that is perpendicular to the surface at the point of incidence. If you are given an angle relative to the surface in your problem statement, subtract this from 90 degrees first and enter the result here.

Why does a calculator sometimes not give an angle of refraction?

This is total internal reflection. When a light ray goes from a denser medium into a less dense one and the angle of incidence exceeds the critical angle, Snell's law requires a sine value that is greater than 1, which has no solution. No light is refracted, so there is no measurable angle, and the calculator marks this situation instead.

What is critical angle?

It is the angle of incidence at which light will travel along the interface, with an angle of 90 degrees. It corresponds to the arcsine of n2 divided by n1 and only exists if the density of the first medium is greater than that of the second. If this angle is exceeded all light rays are reflected internally.

Does light change frequency when it enters a new medium?

No. The frequency is determined by the light source and remains constant on both sides of the interface. Speed and wavelength change by a factor of the index of refraction, which is why the wavelength is shorter in a denser medium.

Can I determine the unknown refractive index with it?

Yes, it is possible. Enter the angle of incidence, the angle of refraction and a known index of refraction and leave the other index blank. The calculator will then re-arrange Snell's law to solve for that index of refraction. Refractometers determine materials in exactly this way.

Angle of Refraction Calculator

Light and the two media

Refractive index of medium 1 (incidence side)

Angle of incidence

Refractive index of medium 2 (refraction side)

Angle of refraction

Show the wavelength in each medium

Enter a vacuum wavelength to see how it shortens inside each material.

More results

Angle of refraction

deg (°)

Ray deviation: deg (°)

Light speed in medium 1 (share of c): %
Light speed in medium 2 (share of c): %

Medium 2 is optically denser, so the ray slows and bends toward the normal. The refraction angle is smaller than the incidence angle.

Frequency stays the same across the boundary, so the wavelength shrinks in step with the speed. Light travels at

75%

of its vacuum speed inside medium 2.

Good to know

Snell's law ties the two media together:

n₁ sin(θ₁) = n₂ sin(θ₂)

. Rearranged, the refraction angle is arcsin(n₁ sin(θ₁) ÷ n₂). Angles are always taken from the normal, the line perpendicular to the surface.

Refraction is why a straw looks bent in a glass of water, why a pool looks shallower than it is, and how lenses, prisms, and fiber optic cables steer light. Diamond's very high index bends light so sharply that it sparkles.

When light passes from one transparent medium into another transparent medium it changes its speed of propagation and is refracted at the interface between them. The angle of refraction is the angle between the refracted ray and the normal, which is a line perpendicular to the interface. This calculator computes this angle using Snell's law and can also be used in reverse.

Enter the index of refraction for each medium as well as the angle of incidence. The calculator will then return the angle of refraction. Alternatively, you can leave one of the four values blank and the calculator will solve for that value, allowing it to identify unknown media or complete missing angles.

What is the angle of refraction?

The reason for refraction is that light travels at different speeds through different media. When a ray of light enters a denser medium it slows down and bends towards the normal. If it enters a less dense medium it speeds up and bends away from the normal.

Both the angle of incidence and the angle of refraction are measured relative to the normal, never relative to the interface itself. If the problem gives you an angle that is measured relative to the interface, subtract it from 90 degrees first.

Snell's Law and the formula for the angle of refraction.

Snell's law relates two indices of refraction and two angles. Each medium has an index of refraction, a dimensionless number equal to the speed of light in vacuum divided by the speed of light in that medium:

n_1 \sin\theta_1 = n_2 \sin\theta_2

The formula can be rearranged to include the angle of refraction which leads to the form that this calculator uses by default:

\theta_2 = \arcsin\!\left(\frac{n_1 \sin\theta_1}{n_2}\right)

The same relationship can be used to calculate other unknown quantities. To determine an angle of incidence or one of the indices of refraction, use this rearranged formula:

\theta_1 = \arcsin\!\left(\frac{n_2 \sin\theta_2}{n_1}\right) \qquad n_1 = \frac{n_2 \sin\theta_2}{\sin\theta_1} \qquad n_2 = \frac{n_1 \sin\theta_1}{\sin\theta_2}

Refractive indices of common media

The index of refraction determines how much a medium will bend light rays. The following values are typical for visible light and can vary slightly depending on wavelength and temperature.

Material	Refractive index
Vacuum	1.0000
Air	1.0003
Ice	1.31
Water (20 C)	1.333
Ethanol	1.36
Fused quartz	1.46
Acrylic (PMMA)	1.49
Crown glass	1.52
Flint glass	1.62
Sapphire	1.77
Diamond	2.42

A full example.

A light ray enters water at an angle of 30 degrees to the normal from air. The refractive index for air is approximately 1.0003 and the refractive index for water is 1.333. Use these values in Snell's law to find the angle of refraction:

\sin\theta_2 = \frac{1.0003 \times \sin 30^\circ}{1.333} = \frac{0.5002}{1.333} \approx 0.3752

\theta_2 = \arcsin(0.3752) \approx 22.03^\circ

The light ray is deflected from the normal and decreases from 30 degrees to about 22 degrees because water slows down light more than air does. The light ray is deflected by about 8 degrees at the interface.

Critical Angle and Total Internal Reflection

When a light ray enters from a denser medium to a rarer medium in the opposite direction, the angle of refraction is greater than the angle of incidence. When the angle of incidence becomes large enough, the refracted ray has to take on a value more than 90 degrees which is not possible.

The critical angle is this threshold at which the refracted ray runs exactly along the surface and forms an angle of 90 degrees:

\theta_c = \arcsin\!\left(\frac{n_2}{n_1}\right), \qquad n_1 > n_2

Once the critical angle is exceeded, no light can escape. All of the light rays are reflected internally, which is called total internal reflection. Fiber optic cables use this principle to keep the ray of light inside and transmit it over many miles. The critical angle at the interface between glass and air is about 41 degrees.

Speed and wavelength in a medium

The refraction is caused by a change in speed. Light that travels at the speed c in vacuum slows down to c divided by the refractive index in a medium. Thus light in water with a refractive index of 1.333 has about 75% of its speed in vacuum. The frequency stays constant, so the wavelength shortens accordingly:

v = \frac{c}{n}, \qquad \lambda = \frac{\lambda_0}{n}

The Wavelength option shows how the vacuum wavelength of light (e.g., green at 500nm) is shortened in each medium.

How to use this calculator

Start with the most common cases. Enter the refractive index of the first medium, the incident angle and the refractive index of the second medium and leave the refraction angle field blank. The calculator will fill in the refraction angle and provide the deviation angle as well as the speed of light in each medium.

To determine a particular material type, enter two angles and one known index of refraction, leaving the other index of refraction free. If light is entering from a denser to a less dense medium, then the result will also show the critical angle. The calculator will give an explicit message if the incident angle causes total reflection.

Where does the phenomenon of refraction occur?

Refraction explains why a straw looks bent when it is placed in water and why a swimming pool appears shallower than it really is. Glasses, camera lenses, prisms as well as endoscopes and microscopes used in medicine work on the principle of refraction.

Engineers use refraction when developing fiber optic networks while gemologists measure the refractive index of precious stones to distinguish diamonds from glass. A simple formula can explain it all.

This tool is for learning and quick estimates only, for accurate optical calculations the actual measured indices of refraction at the particular wavelength and temperature should be used.

Frequently asked questions

Is the angle measured from the edge or normal?: To measure the normal, construct a line that is perpendicular to the surface at the point of incidence. If you are given an angle relative to the surface in your problem statement, subtract this from 90 degrees first and enter the result here.
Why does a calculator sometimes not give an angle of refraction?: This is total internal reflection. When a light ray goes from a denser medium into a less dense one and the angle of incidence exceeds the critical angle, Snell's law requires a sine value that is greater than 1, which has no solution. No light is refracted, so there is no measurable angle, and the calculator marks this situation instead.
What is critical angle?: It is the angle of incidence at which light will travel along the interface, with an angle of 90 degrees. It corresponds to the arcsine of n2 divided by n1 and only exists if the density of the first medium is greater than that of the second. If this angle is exceeded all light rays are reflected internally.
Does light change frequency when it enters a new medium?: No. The frequency is determined by the light source and remains constant on both sides of the interface. Speed and wavelength change by a factor of the index of refraction, which is why the wavelength is shorter in a denser medium.
Can I determine the unknown refractive index with it?: Yes, it is possible. Enter the angle of incidence, the angle of refraction and a known index of refraction and leave the other index blank. The calculator will then re-arrange Snell's law to solve for that index of refraction. Refractometers determine materials in exactly this way.

Related calculators

Snell's Law CalculatorApply Snell's law of refraction to find the angle of refraction, the angle of incidence, or either refractive index, and solve for any of the four. Adds the ray deviation, critical angle and total internal reflection, the speed of light and wavelength in each medium, and Brewster's angle.

Angle of Incidence CalculatorFind the angle of incidence from the angle of refraction and two refractive indices, and solve Snell's law for any of the four quantities. Adds the law of reflection and grazing angle, the ray deviation, the critical angle and total internal reflection, the speed of light and wavelength in each medium, and Brewster's angle.

Brewster Angle CalculatorFind Brewster's angle from the refractive index on each side of a surface, or solve backwards for an unknown index from a measured polarizing angle. Includes the reflectance of each polarization, the 90 degree ray geometry, the critical angle, and a pile-of-plates polarizer model.

Bank Angle CalculatorFind the banking angle of a road or the bank angle of an aircraft turn from the speed and radius, or solve for the speed or radius. Includes the safe speed range with friction, load factor and rate of turn.

Angle of Impact CalculatorFind the angle of impact of a bloodstain from its width and length, estimate the height of origin, and read the spatter shape. Solve for any missing value.

Sun Angle CalculatorFind the sun's angle for any latitude, date, and time. Get the solar altitude, zenith, and azimuth, plus declination, shadow length, and a solar-panel tilt.

Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

HyperPhysics: Refraction and Snell's Law
Georgia State University reference on refraction and index of refraction.
Encyclopedia Britannica: Refraction
Overview of refraction, Snell's law, and the refractive index.
NASA: Refraction of Light
How electromagnetic waves bend when they change media.