Critical Angle Calculator

The Critical Angle Calculator determines the minimum angle of incidence at which total internal reflection (TIR) occurs when light travels from a denser optical medium into a less dense one. Total internal reflection is one of the most important phenomena in geometric optics, underpinning the operation of optical fibers, prisms, binoculars, and diamond cutting.

When a light ray passes from a medium with a higher refractive index (n₁) to one with a lower refractive index (n₂), it bends away from the normal. As the angle of incidence increases, the refracted ray bends further until it travels along the interface at exactly 90° to the normal. The angle of incidence at which this occurs is the critical angle θ_c. For any angle greater than θ_c, all light is reflected back into the denser medium—no refraction occurs.

The critical angle is derived directly from Snell’s law. Setting the refraction angle to 90°, we obtain $$\theta_c = \arcsin\!\left(\frac{n_2}{n_1}\right)$$ where n₁ is the refractive index of the incident medium and n₂ is that of the less dense medium. The critical angle exists only when n₁ > n₂; if the light enters a denser medium, total internal reflection cannot occur.

Optical fibers exploit TIR to transmit light over kilometers with minimal loss. The core of a fiber has a higher refractive index than the surrounding cladding, so light launched at angles exceeding the critical angle bounces along the fiber indefinitely. Similarly, diamonds are cut so that light entering the stone strikes internal facets above the critical angle (~24.4° for diamond–air), maximizing brilliance.

In underwater photography and marine biology, the critical angle for water–air (~48.6°) creates the bright circle known as Snell’s window, through which an underwater observer sees the entire above-water hemisphere compressed into a cone. Understanding this angle is essential for designing aquatic lighting systems and underwater cameras.

This calculator instantly computes the critical angle in both radians and degrees. It also checks whether total internal reflection is physically possible for your chosen pair of media, ensuring you always get a meaningful result.