Critical Angle Calculator
The critical angle is the angle of incidence above which total internal reflection occurs when light passes from a denser medium (higher refractive index n1) to a less dense medium (lower refractive index n2). The formula is theta_c = arcsin(n2/n1), derived by setting the refracted angle to 90 degrees in Snell's law n1 sin(theta1) = n2 sin(theta2). For angles of incidence greater than theta_c, the interface acts as a perfect mirror. This phenomenon underlies optical fiber data transmission, where light is guided along the fiber core by repeated total internal reflections. It is also why gems like diamonds are cut at specific angles to maximize sparkle through internal reflection. Enter the refractive indices to calculate the critical angle in degrees.
Critical angle formula
theta_c = arcsin(n2 / n1)
Derived from Snell's law: n1 sin(theta1) = n2 sin(theta2) with theta2 = 90 degrees. Total internal reflection requires n1 > n2.
Applications of total internal reflection
- Optical fibers: glass core (n ~1.46) with silica cladding (n ~1.44) keeps light confined by TIR.
- Diamonds are cut so that light entering the top undergoes multiple total internal reflections before exiting, maximizing brilliance.
- Retroreflectors (prismatic reflectors in road signs) use TIR in glass prisms at the 45-degree face.
- Medical endoscopes use fiber bundles where TIR transmits images from inside the body.
Critical angle: frequently asked questions
What is the critical angle for total internal reflection?
The critical angle is the minimum angle of incidence (measured from the normal) at which light traveling in a denser medium hits the interface with a less dense medium and undergoes total internal reflection. At angles greater than or equal to the critical angle, no refracted ray exists and all light is reflected back into the denser medium.
What is the formula for the critical angle?
The critical angle theta_c = arcsin(n2/n1), where n1 is the refractive index of the denser medium (where light is traveling) and n2 is the refractive index of the less dense medium (n2 < n1). This comes from setting the refraction angle to 90 degrees in Snell's law.
When does total internal reflection occur?
Total internal reflection requires two conditions: light must be traveling from a denser medium (higher n) to a less dense medium (lower n), and the angle of incidence must equal or exceed the critical angle. Common examples are optical fibers (glass core with lower-index cladding) and diamonds cut to maximize internal reflection.
What is the critical angle for glass in air?
For crown glass (n1 = 1.52) in air (n2 = 1.0), the critical angle is arcsin(1/1.52) = approximately 41.1 degrees. For diamond (n = 2.42) in air, the critical angle is approximately 24.4 degrees, which is why diamonds sparkle brilliantly.
How are optical fibers related to the critical angle?
Optical fibers work by keeping light rays within a glass or plastic core by ensuring they always strike the core-cladding interface at angles greater than the critical angle. The numerical aperture of the fiber is related to the critical angle: NA = sqrt(n1^2 - n2^2), where n1 is the core index and n2 is the cladding index.
Official sources
- OpenStax University Physics Volume 3, Chapter 1: The Nature of Light. openstax.org.
- NIST, Refractive Index of Selected Materials. physics.nist.gov.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.