Critical Angle: Simple Definition, Formula & 4 Worked Examples

The critical angle is the angle of incidence in a denser medium at which the refracted ray runs exactly along the boundary — at 90 degrees to the normal. Any angle larger than this triggers total internal reflection instead of refraction. Here is how to calculate it, plus 4 worked examples for different materials.

Think of θc as the maximum angle at which light can still escape from a medium. Below it, some light refracts out and some reflects. At the threshold, the refracted ray skims the surface. Above it, the light is trapped inside — it cannot escape. This principle makes fibre optics, diamond sparkle, and mirages possible.

What is a critical angle?

It is the threshold that separates ordinary refraction from total internal reflection. When light travelling through a denser medium (higher refractive index) reaches the boundary with a rarer medium (lower refractive index), three things can happen:

• Below θc: light refracts into the rarer medium, bending away from the normal, and some reflects back.
• At θc: the refracted ray bends to exactly 90 degrees — it travels along the boundary surface.
• Above θc: no refraction occurs. All light reflects back into the denser medium by total internal reflection.

This angle exists only when light travels from a higher-index medium to a lower-index medium. Light going from air to water, for example, cannot undergo total internal reflection, so the concept does not apply for that direction.

Critical angle formula

The formula comes directly from Snell's law. At θc, the angle of refraction is 90 degrees, and Snell's law simplifies to:

n₁ sin θc = n₂ sin 90°

Since sin 90° = 1, this becomes:

sin θc = n₂ / n₁

θc = sin⁻¹ (n₂ / n₁)

Where:

• θc is the critical angle
• n₁ is the refractive index of the denser medium (first medium)
• n₂ is the refractive index of the rarer medium (second medium)

For most practical cases, the rarer medium is air with n₂ ≈ 1.00 (actually 1.0003, but 1.00 is close enough for calculations).

The BBC Bitesize guide on the critical angle shows the derivation step by step with clear diagrams.

How to find the critical angle

Finding it is a three-step process:

Step 1. Identify the refractive indices of both media. You need n₁ (the denser medium with higher n) and n₂ (the rarer medium with lower n). Common values: crown glass 1.52, water 1.33, ice 1.31, diamond 2.42, air 1.00.

Step 2. Divide the smaller index by the larger index: n₂ / n₁. This gives you sin θc.

Step 3. Take the inverse sine of the result: θc = sin⁻¹ (n₂ / n₁).

The HyperPhysics calculator lets you plug in different indices to see how θc changes for various materials.

4 worked examples

Example 1: Water to air

Water has a refractive index of n₁ = 1.33. Air is n₂ = 1.00.

sin θc = 1.00 / 1.33 = 0.752 θc = sin⁻¹ (0.752) = 48.6 degrees

This is the most commonly quoted value. Any light ray inside a pool or fish tank that hits the surface at more than 48.6 degrees from the vertical reflects back down. This is why a submerged object viewed from above disappears when you look from a steep angle — the light undergoes TIR at the water surface and never reaches your eye.

Example 2: Crown glass to air

Crown glass has n₁ = 1.52. Air is n₂ = 1.00.

sin θc = 1.00 / 1.52 = 0.658 θc = sin⁻¹ (0.658) = 41.1 degrees

The more precise textbook value is 41.8 degrees (using the exact index). This low threshold is why glass prisms in binoculars and periscopes work so well — light entering the prism hits the internal face at 45 degrees, which is above θc, so it undergoes TIR and reflects 100% of the light without needing a mirrored coating.

Example 3: Diamond to air

Diamond has n₁ = 2.42. Air is n₂ = 1.00.

sin θc = 1.00 / 2.42 = 0.413 θc = sin⁻¹ (0.413) = 24.4 degrees

Diamond's tiny threshold is why it sparkles. Light that enters a diamond hits an internal facet at an angle almost certain to exceed 24.4 degrees, so it undergoes TIR and bounces around inside. A well-cut diamond is designed to keep this light trapped until it exits through the top facets, creating the brilliant flashes. The Gemological Institute of America's guide on diamond sparkle explains how the cut geometry interacts with this principle.

Example 4: Ice to air

Ice has n₁ = 1.31. Air is n₂ = 1.00.

sin θc = 1.00 / 1.31 = 0.763 θc = sin⁻¹ (0.763) = 49.8 degrees

Ice has a slightly lower refractive index than water, so its threshold is a bit higher. This is relevant for understanding how light behaves in ice crystals in the atmosphere — the 49.8-degree value is part of the physics behind halos and sundogs.

Critical angle in everyday life

It is not just a textbook formula — it is the hidden parameter behind several optical phenomena you see every day:

Fibre optics. The core of an optical fibre has a higher refractive index than the cladding. Light injected at an angle above θc bounces along the core by repeated TIR for kilometres with minimal loss.

Diamonds. Diamond's high refractive index (2.42) gives it the smallest threshold (24.4°) of any common gemstone. This traps light effectively, producing the intense sparkle that makes diamonds valuable.

Mirages. On a hot road, the air near the ground is less dense (lower n) than the air above. When light from the sky passes through this temperature gradient at a shallow enough angle, θc is exceeded and TIR occurs, bending the light back upward and creating the illusion of water.

Binoculars and periscopes. Right-angle prisms use the 41.8-degree threshold of glass to reflect light by 90 degrees with 100% efficiency — no metal coating needed.

For a detailed list of how reflection works in these devices, see our guide on examples of reflection of light. The physics of why light bends at boundaries in the first place is covered in what causes refraction.

Common misconception: "the critical angle is the same for all materials"

Many people assume every transparent material has the same value. It actually depends entirely on the refractive index of the material. A high-index material like diamond (n = 2.42) has a small threshold (24.4°), while a low-index material like water (n = 1.33) has a large one (48.6°). Ice (n = 1.31) has an even higher value of about 49.8°.

The relationship is inverse: the higher the refractive index, the smaller this threshold, and the more effectively the material traps light. This is why diamond sparkles more than cubic zirconia (n ≈ 2.15), which sparkles more than glass.

For a detailed comparison of refraction and reflection phenomena, see our guide on reflection vs refraction.