Angle of Refraction Calculator
Apply Snell's law — n₁·sin θ₁ = n₂·sin θ₂. All angles are measured from the normal (perpendicular to the surface), in degrees. Results to 4 decimals.
Refractive index reference
Absolute indices at the sodium D line (verified against published tables).
| Material | n |
|---|---|
| Vacuum | 1.0000 |
| Air (STP) | 1.0003 |
| Ice | 1.3100 |
| Water (20 °C) | 1.3330 |
| Ethanol | 1.3600 |
| Fused quartz | 1.4600 |
| Crown glass | 1.5200 |
| Flint glass | 1.6200 |
| Sapphire | 1.7700 |
| Diamond | 2.4170 |
Frequently asked questions
How do you calculate the angle of refraction?
Use Snell's law: n1 * sin(theta1) = n2 * sin(theta2). Divide the incidence term by the second index and take the arcsine: theta2 = arcsin((n1/n2) * sin(theta1)). Example: light entering water (n = 1.333) from air at a 45-degree incidence angle refracts to 32.0475 degrees.
Is the angle of refraction measured from the normal or the surface?
Angles in Snell's law are always measured from the normal (the line perpendicular to the surface), not from the surface itself. The angle between the ray and the surface equals 90 degrees minus the angle used in the formula. This calculator uses the normal convention throughout.
What is the critical angle and when does it exist?
The critical angle is theta_c = arcsin(n2/n1), the incidence angle whose refracted ray runs exactly along the surface (theta2 = 90 degrees). It exists only when light travels from a denser to a rarer medium (n1 > n2). Examples: crown glass to air = 41.1545 degrees; water to air = 48.6261 degrees; diamond to air = 24.4476 degrees.
What is total internal reflection?
Total internal reflection happens when Snell's law demands sin(theta2) greater than 1, which is impossible, so no refracted ray exists and all light reflects back into the dense medium. It requires n1 > n2 and an incidence angle beyond the critical angle. Example: a 50-degree ray inside crown glass (n = 1.52) hitting air gives sin(theta2) = 1.1640, so the light is fully reflected.
How do you find the refractive index from measured angles?
Rearrange Snell's law: n2 = n1 * sin(theta1) / sin(theta2). Example: with theta1 = 45 degrees in air (n = 1.0003) and a measured theta2 = 32.0475 degrees, n2 = 1.0003*sin(45)/sin(32.0475) = 1.333, the refractive index of water. Both angles must be measured from the normal and greater than 0 degrees.
Why is the refracted ray bent toward the normal when entering a denser medium?
Because sin(theta2) = (n1/n2) * sin(theta1): when n2 > n1, the factor n1/n2 is less than 1, so theta2 is smaller than theta1 and the ray bends toward the normal. Light also slows down in the denser medium (v = c/n). Entering a rarer medium reverses this: the ray bends away from the normal.
What are the refractive indices of common materials?
Absolute refractive indices (sodium D line): vacuum 1.0000, air 1.0003, ice 1.31, water (20 degC) 1.333, ethanol 1.36, fused quartz 1.46, crown glass 1.52, flint glass 1.62, sapphire 1.77, diamond 2.417. This calculator's material picker uses these values.
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