Angle Of Refraction Calculator - Snell's Law Refraction Solver

An angle of refraction calculator is a Snell's law solver that finds the angle a light ray makes with the surface normal after it crosses into a new medium. The result is the angle of refraction, the second of the two angles that define how light bends at a boundary.

• • Physics homework and lab checks: Confirm a textbook or lab result without redoing the Snell's law algebra by hand.
• • Optics and lens prototyping: Compare refraction angles across glass, water, and air when sketching a prism, lens, or window.
• • Fiber optic and waveguide work: Test whether light stays trapped inside a glass core by checking the critical angle for the chosen media.
• • Aquarium and pool sight-line problems: Estimate how much a fish or pool edge appears shifted because of refraction at the water surface.

The angle of refraction is always measured from the surface normal, never from the surface itself. The normal is the imaginary line perpendicular to the boundary at the point where the ray crosses, and every angle in this calculator refers to that line.

If the second medium is optically denser, the refracted angle is smaller than the angle of incidence, so the ray bends toward the normal. If the second medium is less dense, the refracted angle is larger and the ray bends away from the normal.

Because the two angles are defined from the same normal, Angle of Incidence Calculator handles the reverse workflow and reports the incidence, reflection, and critical angles from a single surface angle.

The calculator applies Snell's law in degrees, then checks the result against the critical angle to flag total internal reflection. Three inputs feed one formula, and the result panel reports the refracted angle, the critical angle when it exists, and whether the ray transmits at all.

• θ₁ (angle of incidence): Angle between the incoming ray and the surface normal, in degrees. Entered as the angle of incidence field.
• n₁ (refractive index of medium 1): Refractive index of the medium the light is leaving. Air is 1.000273 at STP.
• n₂ (refractive index of medium 2): Refractive index of the medium the light is entering. Water is 1.333 at 20 °C.
• θ₂ (angle of refraction): Angle between the refracted ray and the surface normal, in degrees. The main output.
• θ_c (critical angle): Smallest angle of incidence that produces total internal reflection, in degrees. Defined only when n₁ > n₂.

The arithmetic happens in radians inside the formula, then converts back to degrees for display, so the inputs and outputs stay in the units a physics class uses. When the ratio (n₁ · sin θ₁) / n₂ exceeds 1, no real angle of refraction exists: the calculator reports no refracted angle, sets the total internal reflection flag to Yes, and shows the critical angle so you can see how far past it the incidence sits.

According to Wikipedia (Snell's law), the law of refraction states n1 sin θ1 = n2 sin θ2, where n1 and n2 are the refractive indices of the two media and θ1 and θ2 are the angles measured from the normal.

According to Wikipedia (Refractive index), the angle of refraction is found by rearranging Snell's law to θ2 = arcsin((n1 / n2) · sin θ1) and the critical angle is arcsin(n2 / n1) when n1 is greater than n2.

Snell's law applied to a curved surface becomes the lens equation, so Thin Lens Equation Calculator uses the same refractive index idea to find object distance, image distance, and focal length for a thin lens.

Four ideas do almost all of the work behind this angle of refraction calculator. Get them straight and the result panel makes sense for any pair of media.

These four ideas are the minimum needed to read any Snell's law problem. Once they are familiar, switching media or chasing a fiber optic design only changes the numbers, not the reasoning.

Curved refracting surfaces need a curvature correction that pure Snell's law cannot supply, so Lensmaker's Equation Calculator uses the same refractive indices together with surface radii to give the focal length of a real lens.

Three numeric inputs and one optional preset, with four outputs. The default example loads with light entering water from air at a 45° angle of incidence, the most common classroom setup.

1. 1 Enter the angle of incidence: Type the angle between the incoming ray and the surface normal, in degrees. The default of 45° is the standard air-to-water example.
2. 2 Set the refractive index of medium 1: Enter n₁ for the medium the light is leaving. Air is 1.000273, water is 1.333, and crown glass is about 1.52.
3. 3 Set the refractive index of medium 2: Enter n₂ for the medium the light is entering. The default of 1.333 is water at 20 °C.
4. 4 Use a preset to fill the indices: Pick a media pair from the preset list to overwrite n₁ and n₂ with the common values for air, water, glass, and diamond.
5. 5 Read the angle of refraction: The first row of the result panel shows the refracted angle in degrees, updated as you type.
6. 6 Check the critical angle and reflection flag: When n₁ is greater than n₂, the panel also shows the critical angle and a Yes/No total internal reflection flag.

A student measures a laser aimed at a glass block at a 30° angle of incidence. They leave the angle field at 30, switch the preset to Air → Crown Glass, and the result panel reads 19.21° for the angle of refraction, no critical angle, and total internal reflection set to No because the light is moving from a less dense medium into a denser one.

The reflected ray produced at the boundary feeds into the mirror equation, so Mirror Equation Calculator applies the same normal-based geometry to find where the reflected image forms in a concave, convex, or plane mirror.

The calculator collapses Snell's law and the critical angle into one result panel. These are the practical reasons to reach for it instead of working the algebra by hand.

• • Fast homework and lab checks: Returns the refracted angle in degrees the moment the inputs change, so the algebra only needs to be done once for understanding, not every time.
• • Catches common unit mistakes: Removes the most common error of mixing degrees and radians, or measuring from the surface instead of the normal, by taking inputs in the units students use.
• • Shows total internal reflection at a glance: The Yes/No flag turns the critical angle from a separate computation into a single readout that you can compare against the angle of incidence.
• • Compare media side by side: Swap n₂ between water, glass, and diamond to see how the refracted angle shifts without re-entering the inputs from scratch.
• • Pairs with related optics tools: The result feeds directly into the thin-lens equation, the lensmaker's equation, and the mirror equation, so this calculator anchors a wider optics workflow.

The calculator is built for quick checks and teaching the geometry of refraction. For deeper modeling of anisotropic crystals, birefringent media, or graded-index fibers, the law still applies but the inputs and outputs need more than the three fields here.

Refraction changes the speed of the wave front rather than just the ray angle, so Harmonic Wave Equation Calculator uses the same refractive index idea to find the speed, wavelength, and frequency of a wave in the new medium.

Snell's law is exact for isotropic media at a fixed wavelength, but real materials and measurement choices can shift the answer by a degree or more.

• • Snell's law assumes isotropic media and a flat interface, so anisotropic crystals such as calcite or quartz need a separate treatment of the ordinary and extraordinary rays.
• • The result panel reports geometric angles but does not split the incoming light into reflected and transmitted intensities, so for absolute reflectance or transmittance use Fresnel's equations instead.

For most textbook and lab work these caveats are small. When the ray grazes the boundary near 90° the result approaches 90° in the second medium, but such rays scatter and stop behaving like clean geometric lines.

According to Britannica, refraction is the change in direction of a wave as it passes from one medium to another, and Snell's law relates the angles to the refractive indices of the two media.