How this instrument works
The angle of refraction is the angle the transmitted ray makes with the normal — the line perpendicular to the boundary, not the boundary itself — once it has crossed into a new medium. Snell's law ties it to the angle of incidence through the two refractive indices: n₁ sinθ₁ = n₂ sinθ₂. Solved for the unknown angle, that becomes θ₂ = asin(n₁sinθ₁ ⁄ n₂). Each index is a stand-in for how much its medium slows light down, since n = c ⁄ v; a higher index means a slower wave, so a ray bends toward the normal entering a denser medium and away from it leaving one.
The sine, not the angle itself, is what the two media trade in equal parts, and that shape comes straight out of the physics at the boundary. Huygens pictured a wavefront as a marching line that wheels around because the edge reaching slower ground pivots while the other edge is still crossing at full speed — exactly the geometry that produces a sine ratio. Fermat's principle of least time reaches the identical formula from a different direction: differentiate the travel time across both media with respect to the crossing point, set it to zero, and sinθ₁ ⁄ v₁ = sinθ₂ ⁄ v₂ falls out on its own.
The formula has a hard edge built into it. Sine cannot exceed 1, so once n₁sinθ₁ is larger than n₂ there is no real θ₂ — the light cannot refract out at all and instead reflects entirely back into the medium it came from. That's total internal reflection, the effect that keeps a signal traveling down an optical fiber core instead of leaking out through the cladding, and it only ever happens on the way from a denser medium into a thinner one.
- Enter the Incident medium refractive index (n₁) — 1.0 for air, 1.33 for water, roughly 1.5 for window glass.
- Enter the Angle of incidence (θ₁) in degrees, measured from the normal to the surface, not from the surface itself.
- Enter the Refractive medium index (n₂) for whatever the ray is entering — 1.5 for crown glass, 2.42 for diamond.
- Read the Angle of refraction (θ₂). If it reports total internal reflection instead, n₁sinθ₁ exceeded n₂ and no refracted ray exists at that angle.
Worked example — sunlight entering window glass
Sunlight strikes a pane of window glass (n₂ = 1.5) from air (n₁ = 1.0) at a 30° angle of incidence, measured from the normal. Snell's law gives sinθ₂ = (1.0 × sin 30°) ⁄ 1.5 = 0.5 ⁄ 1.5 = 0.333333, so θ₂ = asin(0.333333) = 19.47° — the figure the Angle of refraction field returns for these three inputs.
That 10.5° of bending toward the normal is the same effect that makes a pencil in a glass of water look snapped at the surface. Push the incidence angle to a full 90° — light skimming the glass edge-on — and θ₂ tops out at asin(1 ⁄ 1.5) = 41.81°, the steepest angle this particular glass can ever bend an incoming ray to on its way in.
Questions
Why is the angle measured from the normal instead of the surface?
Because the normal is the one reference line that stays perpendicular to the boundary regardless of how that boundary is oriented, so angles measured from it are unambiguous and match the sine relationship Snell's law is built on. Measuring from the surface would bury a 90° offset inside every calculation.
What happens when n1 sinθ1 works out larger than n2?
There is no real solution, since sine cannot exceed 1. Physically that is total internal reflection: past a critical angle θc = asin(n2 ⁄ n1), the entire ray reflects back into its starting medium instead of refracting out — the mechanism that keeps light trapped inside an optical fiber core.
Does the angle of refraction change with the color of the light?
Yes. Refractive index depends on wavelength, an effect called dispersion, so violet light bends slightly more than red light crossing the same boundary because it sees a marginally higher n. This calculator's single n₂ input assumes one wavelength; a glass prism spreads white light into a spectrum precisely because dispersion means it can't.
Who actually calculates an angle of refraction?
An optical engineer grinding a lens or prism toward a target focal length, a gemologist reading a refractometer to separate a real sapphire from glass, and a fiber-optic engineer checking that light launched into a cable core stays under the critical angle so it reflects down the fiber rather than leaking out through the cladding.
Why does light bend toward the normal entering glass but away from it leaving glass?
The direction follows the speed change, not the medium's name. Entering the higher-index medium, light slows and the wavefront pivots toward the normal; leaving into the lower-index medium it speeds back up and pivots away. Reverse the 30° air-to-glass example above and the same ray exits glass into air at 48.59° instead of 19.47°.
Can the angle of refraction be larger than the angle of incidence?
Yes, whenever n2 is smaller than n1. Light leaving a denser medium for a thinner one always refracts to a wider angle than it arrived at, right up to the critical angle, beyond which it stops refracting altogether and reflects internally instead.