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Critical Angle Calculator

Calculate the critical angle for total internal reflection between two optical media.

Result

Critical Angle
41.81°

At any angle of incidence beyond this critical angle, light traveling from the denser to the less dense medium undergoes total internal reflection instead of refracting through.

About the Critical Angle Calculator

This calculator finds the critical angle at which light traveling from a denser optical medium into a less dense one stops refracting and instead undergoes total internal reflection. It is a standard tool for optics coursework and for anyone working with fiber optics, prisms, or gemstone cutting where controlling internal reflection matters.

How It Works

You provide the refractive index of the denser medium the light starts in (n1) and the refractive index of the less dense medium it would exit into (n2). The calculator takes the inverse sine of the ratio n2/n1 to find the critical angle in radians, then converts that to degrees. Beyond this angle of incidence, light striking the boundary reflects entirely back into the denser medium rather than passing through.

critical angle = arcsin(n2 / n1), converted from radians to degrees

Formula & Methodology

The underlying physics comes from Snell's Law, n1 sin(theta1) = n2 sin(theta2). The critical angle is the special case where the refracted angle theta2 reaches exactly 90 degrees, meaning sin(theta2) = 1. Substituting that in and solving for theta1 gives sin(theta1) = n2/n1, so theta1 = arcsin(n2/n1). The calculator performs exactly this substitution and reports theta1 in degrees.

Examples

Glass to air boundary

With n1 set to 1.5 (typical glass) and n2 set to 1.0 (air), the critical angle comes out to about 41.81 degrees, meaning any light hitting the glass-air boundary from inside at more than 41.81 degrees from the normal reflects entirely back into the glass.

Diamond to air boundary

Using n1 = 2.42 (diamond) and n2 = 1.0 (air), the critical angle is roughly 24.41 degrees. This small critical angle is a major reason cut diamonds display strong internal light trapping and brilliance.

Advantages

  • Enforces the correct physical ordering of the two refractive indices, preventing a nonsensical negative or undefined result
  • Converts the raw radian output of the arcsine function into degrees automatically, matching how optical angles are typically reported
  • Includes a plain-language note explaining what happens physically at angles beyond the calculated critical angle

Common Mistakes

  • Entering the less dense medium's index as n1 and the denser medium's as n2, which triggers the calculator's built-in error since it requires n1 > n2
  • Confusing the critical angle with the angle of refraction at a specific incidence angle, when it is instead the single threshold angle at which refraction stops entirely
  • Assuming the critical angle formula applies for light traveling from a less dense into a denser medium, where total internal reflection cannot occur at all

Edge Cases to Watch For

  • If either refractive index is entered as zero or negative, the calculator returns an error since refractive indices must be positive physical quantities.
  • If n2 is greater than or equal to n1, the calculator returns an error, because total internal reflection is only geometrically possible when light travels from a denser medium into a less dense one.
  • As n2/n1 approaches 1 (the two media becoming nearly equal in density), the critical angle approaches 90 degrees, meaning total internal reflection becomes achievable only at extremely grazing angles.

Common Use Cases

  • Optics and physics students solving total internal reflection problems for coursework or lab reports
  • Fiber optic and photonics engineers checking that light entering a fiber core will remain trapped by internal reflection
  • Gemologists and jewelry designers reasoning about how cut angles affect a stone's light-trapping brilliance
Written & fact-checked by the Calculateus TeamLast updated August 5, 2026How we verify our formulas

Frequently asked questions

Why does fiber optic cable rely on the critical angle?

Fiber optic cables are designed so light entering the fiber core always hits the core-cladding boundary at an angle beyond the critical angle, guaranteeing total internal reflection - this lets light bounce down the length of the fiber with essentially no loss through the sides, even around gentle bends.

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Conclusion

The critical angle marks the exact tipping point between refraction and total internal reflection, and this calculator derives it directly from the two media's refractive indices. Because the relationship only holds when light moves from denser to less dense material, the tool's built-in ordering check keeps results physically meaningful.