The critical angle is defined as the highest angle of incidence that does not result in total internal reflection; it is the angle at which refraction will occur at pi/2 radians (90 degrees), the light traveling along the boundary between the two media.
By Snell's Law, we know that angle of refraction and refractive index are related as follows:
`n_1 sin (theta_1) = n_2 sin (theta_2)`
At the critical angle, by definition `theta_2 = pi/2`, so `sin(theta_2) = 1`.
`n_1 sin(theta_1) = n_2`
Solving for theta_1 gives us the critical angle in terms of the refractive indices:
`theta_1 = sin^{-1} (n_2 / n_1)`
Notice how this will be undefined if `n_2 gt n_1`; total internal reflection (and thus, a critical angle) only occurs when light travels to a medium with a lower refractive index. All we need now is to know the refractive indices of glass and diamond. Diamond we know quite precisely as 2.417; glass varies a bit depending on its precise composition, but is usually about 1.5.
`theta_1 = sin^{-1} (1.5/2.417) = sin^{-1}(0.62)`
`theta_1 = 0.67 rad = 38 deg`
The critical angle is about 38 degrees.
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