Snell's Law Calculator
Enter the angle of incidence and the refractive indices of the two media to find the angle of refraction from Snell's law. The calculator also reports the critical angle and warns you when total internal reflection occurs.
Find the Angle of Refraction from Snell's Law
Snell's Law Formula
Where:
- \(n_1\) — refractive index of the first medium (the one the light comes from)
- \(\theta_1\) — angle of incidence, measured between the incoming ray and the normal to the surface
- \(n_2\) — refractive index of the second medium (the one the light enters)
- \(\theta_2\) — angle of refraction, measured between the refracted ray and the normal
Solving for the angle of refraction gives \(\theta_2 = \arcsin\!\big(\tfrac{n_1}{n_2}\sin\theta_1\big)\). Both angles are always measured from the normal (the line perpendicular to the surface), not from the surface itself.
When light travels from a denser medium into a less dense one (\(n_1 > n_2\)), a critical angle exists:
\[ \theta_c = \arcsin\!\left(\frac{n_2}{n_1}\right) \]
For any incidence angle greater than \(\theta_c\), Snell's law has no solution and the ray is completely reflected back — this is total internal reflection.
Snell's Law – Calculation Example
Example: a light ray in air hits the flat surface of a pool of water at an angle of 40° to the normal. Air has \(n_1 = 1\) and water has \(n_2 = 1.33\). Find the angle of the refracted ray inside the water.
\[ \sin\theta_2 = \frac{n_1}{n_2}\sin\theta_1 = \frac{1}{1.33} \times \sin 40^\circ = 0.7519 \times 0.6428 = 0.4833 \]
\[ \theta_2 = \arcsin(0.4833) = 28.90^\circ \]
Result: the ray refracts at 28.90° (about 28° 54′, or 0.5044 rad) from the normal. Because water is optically denser than air, the ray bends toward the normal — 40° in, 28.90° out.
How to use the calculator
- Enter the Angle of incidence (θ₁) and pick its unit (degrees or radians) from the dropdown. The angle must be measured from the normal to the surface, between 0° and 90°.
- Enter the Refractive index of medium 1 (n₁) — the medium the light is coming from (for air, 1 is accurate enough).
- Enter the Refractive index of medium 2 (n₂) — the medium the light is entering.
- Read the Angle of refraction (θ₂) in degrees, degrees and minutes, and radians.
- Check the second result for the critical angle and total internal reflection: if it exists, any incidence angle above it is totally internally reflected.
What Snell's law describes
Snell's law (also called the law of refraction) governs how light changes direction when it crosses the boundary between two transparent materials. Light travels at different speeds in different media, and the refractive index \(n\) expresses this: \(n = c/v\), where \(c\) is the speed of light in vacuum and \(v\) is the speed in the material. When a ray passes into a medium with a higher index, it slows down and bends toward the normal; passing into a lower-index medium, it speeds up and bends away from the normal.
The law was discovered experimentally in 1621 by the Dutch astronomer Willebrord Snellius and later derived theoretically, most elegantly from Fermat's principle of least time: light follows the path that minimizes travel time between two points. Snell's law explains familiar effects such as the apparent bending of a straw in a glass of water, the shallowness of pools (objects underwater appear closer to the surface than they really are), and the way lenses focus light in eyeglasses, cameras, and microscopes.
Interpreting the output is straightforward: if \(\theta_2 \theta_1\), it entered a less dense medium and bent away. Note that a ray hitting the surface head-on (\(\theta_1 = 0\)) passes straight through without bending, regardless of the indices.
Critical angle and total internal reflection
When light goes from a denser medium to a less dense one (\(n_1 > n_2\), e.g. water to air or glass to air), the refracted ray bends away from the normal. As \(\theta_1\) grows, \(\theta_2\) reaches 90° at some point — that incidence angle is the critical angle \(\theta_c\). Beyond it, no refracted ray can exist: all the light is reflected back into the first medium with nearly perfect efficiency. This is total internal reflection.
For a water–air boundary, \(\theta_c = \arcsin(1/1.33) \approx 48.75^\circ\); for common glass–air, it is about 41.8°. Total internal reflection is the working principle of optical fibers that carry internet traffic, of prism binoculars and periscopes, and of the brilliant sparkle of cut diamonds (\(n \approx 2.42\), critical angle only 24.4°).
Typical refractive indices
| Medium | Refractive index (approx., visible light) |
|---|---|
| Vacuum | 1.0000 |
| Air | 1.0003 |
| Ice | 1.31 |
| Water | 1.33 |
| Ethanol | 1.36 |
| Crown glass | 1.52 |
| Sapphire | 1.77 |
| Diamond | 2.42 |
Assumptions and limitations
- The calculation assumes a flat, smooth boundary and monochromatic light. Real materials are dispersive: the index varies slightly with wavelength (blue light bends more than red), which is why prisms split white light into a spectrum.
- Some light is always reflected at the boundary even below the critical angle (typically 4–8% for glass–air at moderate angles); Snell's law describes only the transmitted ray's direction, not the energy split.
- The angles must be measured from the normal. A common mistake is entering the angle from the surface instead — subtract it from 90° before entering.
- Refractive indices must be positive; entering 0 for either medium is physically meaningless and will be rejected.
- The calculator does not model absorption, scattering, or behavior in anisotropic (birefringent) materials such as calcite, where a single refractive index is not well defined.
For more physics tools, see our kinematic equations solver or the conservation of energy calculator.
Frequently Asked Questions
How do I calculate the angle of refraction by hand?
Write down \( n_1 \sin\theta_1 = n_2 \sin\theta_2 \), then rearrange to \( \sin\theta_2 = (n_1/n_2)\sin\theta_1 \). Compute the right-hand side with a sine table or scientific calculator, then take the inverse sine (arcsin) of the result. Make sure your calculator is in degree mode if your angles are in degrees. For example, from air into glass with \(\theta_1 = 30^\circ\): \( \sin\theta_2 = (1/1.52) \times 0.5 \approx 0.329 \), so \( \theta_2 \approx 19.2^\circ \).
Does Snell's law work in reverse, for light going from water or glass into air?
Yes. Refraction is reversible: if a ray bends by a certain amount going from medium A into B, a ray traveling the exact opposite path from B into A retraces it. Simply swap the two indices and the two angles — the same equation holds. The only difference is the direction of bending and the possible onset of total internal reflection when going from the denser to the less dense medium.
Why does the calculator say there is no critical angle for light going from air into glass?
A critical angle only exists when light travels from a denser medium into a less dense one (\( n_1 > n_2 \)), because only then can the refracted angle reach 90° as the incidence angle grows. Going from air into glass, the ray bends toward the normal, so the refracted angle never exceeds the incidence angle and can never reach 90° — total internal reflection is impossible in that direction.
What if I measured the angle from the surface instead of the normal?
Subtract it from 90° before entering it. Snell's law requires angles measured from the normal — the imaginary line perpendicular to the surface. For example, a ray hitting water at 50° to the surface enters at \( 90^\circ - 50^\circ = 40^\circ \) to the normal. This is the single most common source of wrong answers in refraction problems, and it also applies to the output: the result is from the normal too.
Can I use this calculator for sound, water waves, or other waves?
Yes, as long as each medium can be described by a single index (ratio of wave speeds). Snell's law applies to any wave crossing a boundary between two media — sound between air and water, seismic waves between rock layers, or water waves between regions of different depth. Use \( n = v_{ref}/v \) for each medium, where \( v \) is the wave speed, and the same formula applies.
How do I convert between degrees and radians for the angles?
Multiply degrees by \( \pi/180 \) to get radians, and multiply radians by \( 180/\pi \) to get degrees. For instance, \( 40^\circ = 40 \times 0.01745 \approx 0.6981 \) rad, and \( 1 \) rad \( \approx 57.3^\circ \). The calculator accepts either unit for the input angle via the dropdown and reports the results in both, so manual conversion is usually unnecessary.
Does the refractive index change with the color of light?
Yes, slightly — a property called dispersion. The index is higher for blue light than for red in almost all materials, which is why a prism splits white light into a spectrum. The difference is small (a few percent for glass), so using a single tabulated value is fine for most purposes. If you need precision, look up the index at your specific wavelength, such as the sodium D line (589 nm) commonly used in reference tables.