Angle of Incidence Calculator

Calculate the angle of incidence from the angle of refraction and the refractive indices, or solve Snell's law for any quantity. Shows the law of reflection, grazing angle, critical angle, total internal reflection, and Brewster's angle.

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Mathematics

Geometry

Angle of Incidence Calculator

Calculate the angle of incidence from the angle of refraction and the refractive indices, or solve Snell's law for any quantity. Shows the law of reflection, grazing angle, critical angle, total internal reflection, and Brewster's angle.

Angle of Incidence Calculator

Enter any three values

Enter the two refractive indices and the angle of refraction to get the angle of incidence, or fill any three of the four fields and leave the unknown blank. Every angle is measured from the normal, the line perpendicular to the surface, not from the surface itself.

Show the reflected ray

The angle of incidence also fixes the reflected ray: it leaves at the same angle on the other side of the normal (the law of reflection). This layer adds the angle of reflection, the grazing angle measured from the surface, and how far the mirror turns the ray.

Show wave speed and wavelength

Refractive index is the speed of light in a vacuum divided by the speed in the medium, so this layer shows how fast light travels in each medium. Add a vacuum wavelength to see how the wavelength shrinks in each medium too.

Add Brewster's polarizing angle

Brewster's angle is the special angle of incidence at which the reflected light becomes completely polarized. It follows straight from the two refractive indices.

Incidence-refraction curve

Show the incidence-refraction curve

Plot the angle of refraction against the angle of incidence for your two media, so you can watch the curve bend and, past the critical angle, cut off at total internal reflection.

Common materials

Show the common-materials table

Compare the refractive index, the speed of light, and the critical angle of common media, from vacuum and air to water, glass, and diamond, with your own two media added alongside.

Refractive index, speed of light, and critical angle into air for common media, with your own media added

Medium

Refractive index

Speed of light (km/s)

Critical angle to air

Vacuum1299,792-
Air1299,70388.60 deg
Water (20 C)1.333224,90148.61 deg
Ethanol1.361220,27447.29 deg
Crown glass1.52197,23241.14 deg
Sapphire1.77169,37434.40 deg
Diamond2.417124,03524.44 deg
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The angle of incidence is the angle between an incoming light ray and the normal, where the normal is an imaginary line perpendicular to the surface at the point where the light ray hits it. This angle is always measured first when encountering a boundary with either light or any other wave, as it determines how the light will be reflected and refracted. This calculator computes the angle of incidence from the angle of refraction and two indices of refraction. The same relationship can be used in reverse to determine any one of the four values given the other three. Enter any three values and leave the fourth blank.

What is the angle of incidence?

The angle is measured from the normal, not the surface. This often confuses people. A light ray that is almost parallel to the surface has an incident angle of about 90 degrees; a light ray hitting the surface perpendicularly has an incident angle of 0 degrees, which is called the normal case. The angle measured from the surface itself is the scattering angle and it is equal to 90 minus the incident angle.

The same angle determines two possible scenarios that can occur when a ray of light hits an interface. Part of the light is reflected back as the reflected light, while another part crosses the interface and becomes refracted light. The incident angle determines both these phenomena simultaneously.

Law of reflection:

For reflected light there is a very simple rule: the angle of reflection equals the angle of incidence, and the incident ray, the reflected ray, and the normal all lie in the same plane.

θreflection=θincidence\theta_{\text{reflection}} = \theta_{\text{incidence}}

A light ray hitting a flat mirror at an angle of 40 degrees to the normal will leave the mirror on the other side of the normal at an angle of 40 degrees. This is why a mirror angled by 45 degrees can turn a light beam through a right-angle, and it's the principle behind periscopes. By activating the layer for reflected rays you can see the reflection angle, the scattering angle, and the total angle that a ray turns at a given incident angle.

Angle of incidence in refraction

Some of the light that crosses a boundary is deflected because the speed of light varies in different materials. The degree to which a material slows down the speed of light is its index of refraction, denoted by n. The index of refraction of vacuum is exactly 1, air's is slightly higher at about 1.0003, water's is 1.333, normal glass is around 1.52 and diamond is 2.417. Snell's law relates the angle of incidence to the angle of refraction using two indices of refraction:

n1sin⁡(θ1)=n2sin⁡(θ2)n_1 \sin(\theta_1) = n_2 \sin(\theta_2)

where n1 and n2 are the indices of refraction for the first and second medium respectively, θ1 is the angle of incidence, and θ2 is the angle of refraction, with both angles measured from the normal. To solve for the incident angle specifically, we can rearrange the formula as follows:

θ1=arcsin⁡ ⁣(n2sin⁡(θ2)n1)\theta_1 = \arcsin\!\left(\frac{n_2 \sin(\theta_2)}{n_1}\right)

Example calculation

Suppose a light ray enters from air into glass and the angle between the refracted ray and the normal is 19.5 degrees. The index of refraction for air is about 1.0003 while the index of refraction for glass is 1.50. To isolate the unknown incident angle, use:

sin⁡(θ1)=n2sin⁡(θ2)n1=1.50×sin⁡(19.5∘)1.0003≈0.500\sin(\theta_1) = \frac{n_2 \sin(\theta_2)}{n_1} = \frac{1.50 \times \sin(19.5^\circ)}{1.0003} \approx 0.500

Calculate the arcsine of .500; the angle of incidence is approximately 30 degrees. When a ray of light enters a denser medium like glass it slows down and bends by about 10.5 degrees from the normal line when entering. Leave any field blank to calculate in the desired direction.

Solutions for different directions

Since Snell's law is a single equation that relates four quantities, it can be rearranged to solve for any one of the quantities if you know the other three. No modes need to be selected; leaving a field blank tells the calculator which quantity to solve for.

To find

Rearranged formula

In words

Angle of incidence

theta1 = arcsin(n2 sin(theta2) / n1)

From both indices and the angle of refraction

Angle of refraction

theta2 = arcsin(n1 sin(theta1) / n2)

From both indices and the angle of incidence

Index of medium 1

n1 = n2 sin(theta2) / sin(theta1)

From the other index and both angles

Index of medium 2

n2 = n1 sin(theta1) / sin(theta2)

From the other index and both angles

Critical angle and total internal reflection

When a light ray hits a medium with lower index of refraction and higher speed of propagation, such as from glass into air, it is bent away from the normal line. As the angle of incidence increases, the refracted ray bends farther from the border until at some critical angle it runs along the border at 90 degrees. This is called the critical angle. By setting the refraction angle to 90 degrees in Snell's law we get:

θc=arcsin⁡ ⁣(n2n1)\theta_c = \arcsin\!\left(\frac{n_2}{n_1}\right)

When the angle of incidence is greater than the critical angle, light has no way out. It gets reflected back into the first medium completely; this phenomenon is called total internal reflection. This only happens when a ray of light goes from a higher refractive index medium to a lower refractive index medium. That's why this calculator shows the critical angle only if the first medium is denser than the second one. When an angle of incidence greater than the critical angle is entered, the calculator marks it as total internal reflection and does not return any impossible angle of refraction. This explains why light flows in fiber optic cables, diamonds sparkle, and there are shimmering mirages on hot roads.

The brewster angle, a special incident angle.

Another angle of incidence is hidden in the two refractive indices. At Brewster's Angle, the reflected light rays are completely polarized and oscillate in only one direction. Simultaneously, the reflected ray and the broken ray form an exact 90-degree angle. This angle equals the arctangent of the second refractive index divided by the first refractive index. Polarizing sunglasses are designed to reduce this glare that is reflected from water surfaces and roads. Photographers use polarizing filters for the same reason.

The important role of the angle of incidence.

Every lens in a pair of glasses, camera, microscope or telescope is shaped to direct the angles of light hitting different surfaces towards a focus. When sunlight hits at an angle other than straight down, solar cells lose power. Fibre optic cables use total reflection to transmit internet signals. Phenomena such as prisms, rainbows, mirages, swimming pool bottoms appearing shallower and straws in glasses looking bent are all examples of the effects of angles of incidence.

This concept is not limited to light. As it is about the way any wave hits a boundary, the principle of angle of incidence also applies to sound waves, earthquake waves and radio waves, as long as the material has the same properties in all directions.

This tool is for learning purposes, to create homework and solve everyday optical problems. The refractive indices vary slightly with wavelength and temperature. So use values measured for the material and conditions when doing accurate calculations.

Frequently asked questions

What is the angle of incidence?

The angle of incidence is the angle between the incident ray and the normal. The normal is a line that is perpendicular to the surface at the point where the ray hits it. It is measured from the normal, not from the surface. If the light ray hits the surface directly, then the angle of incidence is 0 degrees (direct incidence). If the light ray grazes along the surface almost horizontally, then the angle of incidence approaches 90 degrees.

Is the angle of incidence measured from the surface or from the normal?

The measurement must always be from the normal, which is perpendicular to the surface. This is a common mistake in optics. The angle measured relative to the surface is called the scattering angle and is equal to 90 degrees minus the incident angle. So if the light direction is 30 degrees relative to the normal, it is 60 degrees relative to the surface.

What is the angle of incidence in the law of reflection?

The law of reflection states that the angle of incidence is equal to the angle of reflection, both measured from the normal and in the same plane. A light ray hitting a mirror at an angle of 40 degrees will leave it on the other side of the normal at an angle of 40 degrees. This is why a mirror with a 45-degree opening can turn a light beam by 90 degrees.

How do you find the angle of incidence given the angle of refraction?

Arrange the Snell's law as sin(θ₁) = (n₂*sin(θ₂)) / n₁ and then take the arcsine. For example, a light ray entering glass from air with an angle of refraction of 19.5 degrees (refractive index of glass is 1.50, refractive index of air is 1.0003) would have an incident angle of arcsin((1.50*sin(19.5 degrees)) / 1.0003) which is approximately equal to 30 degrees. In this calculator, you input the two indices of refraction and the angle of refraction, leaving the field for the angle of incidence blank.

What is the angle of incidence for total reflection?

Total internal reflection occurs when the angle of incidence exceeds the critical angle. The critical angle only exists if light is traveling from a medium with a higher index of refraction to one with a lower index of refraction. The critical angle is arcsin(n2/n1). For the interface between glass and air, the critical angle is about 42 degrees. So any ray of light coming from inside the glass that hits the interface at an angle greater than 42 degrees will be totally internally reflected and not refracted out into the air.

What are some typical indices of refraction?

The refractive index for a vacuum is exactly 1, air is about 1.0003, water is 1.333, ethanol is 1.361, ordinary crown glass is about 1.52, sapphire is 1.77 and diamond is 2.417. The higher the refractive index, the slower light travels and the more pronounced the deflection. The list of common materials in this calculator includes these values as well as the corresponding speed of light and critical angle for each material.

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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

  1. Wikipedia: Snell's law

    The law of refraction, how the angle of incidence and the angle of refraction relate, and the critical angle.

  2. Wikipedia: Reflection (physics)

    The law of reflection, specular versus diffuse reflection, and the angle of incidence measured from the normal.

  3. HyperPhysics: Refraction and Snell's Law

    Reference for the angle of incidence, refractive index, the critical angle, and Brewster's angle with worked relations.