Snell's Law, the Critical Angle, and Which Way the Ray Bends

Snell’s law bends a ray; the critical angle says when it stops bending and turns back. This gives the refracted angle, the critical angle for that pair of media in that direction, the Brewster angle, the speed of light in each, and it refuses to answer rather than inventing a refracted ray that does not exist. Every angle is in degrees; the arcsine underneath returns radians and the page converts.

Last updated 11 Nov 2026

Refracted angle, critical angle, Brewster angle

Two indices, one angle, degrees throughout

Refractive index of the medium the light starts inDimensionless, and never less than 1 for an ordinary transparent material. Vacuum is exactly 1; air is 1.000293, which is 1 for every school purpose. NCERT’s Table 9.1 gives water 1.33, crown glass 1.52 and diamond 2.42, and those are the values used in the table on this page, with each one’s stated critical angle reproduced to better than a hundredth of a degree as a check. Index depends slightly on wavelength — that is dispersion, and it is why a prism makes a spectrum — so a single number is a single colour, conventionally the sodium yellow line.

Refractive index of the medium it crosses intoThe second medium. Whether there is a critical angle at all depends on the ORDER of these two fields and not on the pair. Going from glass to air there is one, at 41.14° for crown glass; going from air to glass there is none, at any angle whatever, because the ray bends towards the normal and can never reach 90°. If this field is the larger of the two, the critical-angle row below is deliberately blank.

Angle of incidence, from the normalIn degrees, and measured from the NORMAL — the perpendicular to the surface — not from the surface itself. An angle measured from the surface is the complement of the one you want, so 30° from the surface is 60° of incidence, and the mistake costs a whole answer rather than a sign. Zero means straight in, which refracts not at all whatever the two indices are. The engine’s trigonometry works in radians; this page multiplies by π/180 on the way in and by 180/π on the way out, and prints the radian value as a row so you can see which is which.

49.4642°Example

Light inside crown glass (n = 1.52) reaching an air surface (n = 1.00) at 30° of incidence

One equation, and the two different things it does depending on direction

n₁ sin i = n₂ sin r · i_c = arcsin(n₂/n₁), only when n₁ > n₂ · n = c/v · λ_medium = λ_vacuum/n · i_B = arctan(n₂/n₁)

n₁ sin i = n₂ sin r: Snell’s law. Both angles are measured from the normal, which is the perpendicular to the surface. Measuring from the surface instead gives you the complement and destroys the answer rather than merely shifting it, and it is the commonest error on the topic after radians
into a denser medium: n₂ > n₁, so sin r < sin i and the ray bends TOWARDS the normal. There is no critical angle in this direction at any incidence. Even at a grazing 90° the refracted angle only reaches arcsin(n₁/n₂) — 41.14° for air into crown glass, which is why a glass block has a cone of accessible directions inside it and nothing outside that cone
into a rarer medium: n₁ > n₂, so the ray bends AWAY from the normal and the refracted angle outruns the incident one. At some incidence the refracted angle reaches 90° and beyond it there is nothing: total internal reflection. This is the only direction in which a critical angle exists, and it is why the page’s critical-angle row is sometimes blank
i_c = arcsin(n₂/n₁): the critical angle. It is a property of the PAIR and the DIRECTION, never of one material: crown glass to air is 41.14°, air to crown glass does not exist, and crown glass to water is 61.0°. NCERT’s Table 9.1 gives water 48.75°, crown glass 41.14° and diamond 24.41° against air, and each reproduces from arcsin(1/n) to better than 0.01°
radians: the engine underneath this page works in radians, as every programming language does. asin() returns 0.863 313 for the default inputs, not 49.46°. The page multiplies the incidence by π/180 going in and the arcsine by 180/π coming out, and prints the radian value as its own row so the two cannot be confused. If you are writing this calculation yourself, in a spreadsheet or in any language, the missing conversion is where it will go wrong: 49.46 radians is 2834 degrees
i_B = arctan(n₂/n₁): the Brewster angle, where the reflected light is completely polarised perpendicular to the plane of incidence. 56.66° for air into crown glass. Unlike the critical angle it exists in BOTH directions — arctan is defined for any positive argument — and it is a different phenomenon entirely: polarisation of the reflected beam rather than disappearance of the refracted one. It is why polarising sunglasses cut glare from a road or a lake

Worked example

Light inside crown glass (n = 1.52) reaching an air surface (n = 1.00) at 30° of incidence
CHECK THE DIRECTION BEFORE YOU CALCULATE. n₁ is larger than n₂, so the light is leaving the denser medium: the ray will bend AWAY from the normal and the refracted angle will be bigger than 30°. Knowing that first catches an inverted fraction immediately, and it also tells you that a critical angle exists in this direction.
THE ARITHMETIC. sin r = n₁ sin i / n₂ = 1.52 × sin 30° / 1.00 = 1.52 × 0.5 = 0.76. Then r = arcsin(0.76). In radians that is 0.863 313, and multiplied by 180/π it is 49.4642°. Bigger than 30°, as predicted. The deviation — how far the ray actually turned — is 19.464°.
THE RADIAN TRAP, SPELLED OUT. Every calculator and every programming language returns 0.863 313 from that arcsine. If you report 0.863 degrees you are out by a factor of 57.3; if you feed 30 straight into a sine function expecting degrees you get sin(30 rad) = −0.988, and Snell's law then hands you a negative sine and no answer at all. This page prints the radian value as a row beside the degree value for exactly this reason.
NOW PUSH THE INCIDENCE UP AND WATCH IT RUN OUT. At 35°, sin r = 1.52 × 0.573 58 = 0.871 84 and r = 60.673°. At 40°, sin r = 0.977 04 and r = 77.698°. The refracted ray is racing towards the surface, and at 41.1395° of incidence sin r = 1.52 × 0.657 895 = 1.0000 exactly and the refracted ray lies flat along the boundary. That is the critical angle, arcsin(1/1.52) = 41.139 51°, and it matches the 41.14° printed in NCERT's Table 9.1. Notice how fast it moves at the end: the last two degrees of incidence are worth more than twelve degrees of refraction: 40° gives 77.70° and 41.14° gives 90°.
AND PAST IT, THE PAGE REFUSES. At 50° of incidence in the glass, sin r = 1.52 × sin 50° = 1.164. No angle has a sine greater than 1, so there is no refracted ray: all of the light is reflected back into the glass. That is total internal reflection, and it is the right answer rather than a failure — which is why this page prints no number for it instead of returning something plausible. It is how an optical fibre works, why a diamond sparkles with its 24.41° critical angle, and why a fish sees the whole sky compressed into a 97.5° cone directly above it.
NOW RUN IT BACKWARDS, air into glass, and the critical angle vanishes. sin r = 1.00 × sin 30° / 1.52 = 0.328 947, so r = 19.2049° — smaller than the incidence, bending towards the normal. Push the incidence to a grazing 90° and the refracted angle only reaches arcsin(1/1.52) = 41.14°. It can never get to 90°, so there is no critical angle at all in this direction, and the page leaves that row blank. The same two glasses, the same two numbers, and the phenomenon exists one way and not the other.

Critical angles against air, from NCERT’s own table, each checked by arithmetic

Three rows, each one checked rather than copied: arcsin(1/n) is computed from the index NCERT prints and compared with the critical angle NCERT prints beside it, and all three agree to better than a hundredth of a degree. That check is worth doing because it is the kind of table that gets copied from site to site without it. The table in the printed chapter has a fourth row, dense flint glass, which this batch read as 1.62 with a critical angle of 37.31° — and those two numbers are NOT consistent, since arcsin(1/1.62) is 38.12° and 37.31° needs an index of 1.650. Either the copy read here was garbled or the row is internally inconsistent; this batch could not establish which, so dense flint glass appears nowhere on this page. The last row is the one most tables omit and it is the one students most need: going INTO the denser medium there is no critical angle at all, at any angle of incidence, because the refracted ray bends towards the normal and can never reach 90°.
Medium	Index n	Critical angle to air, arcsin(1/n)	NCERT Table 9.1 prints	Agrees
Water	1.33	48.7535°	48.75°	Yes, to 0.0035°
Crown glass	1.52	41.1395°	41.14°	Yes, to 0.0005°
Diamond	2.42	24.4075°	24.41°	Yes, to 0.0025°
Air into any of the above	—	none exists	—	There is no critical angle going into a denser medium

The critical angle belongs to a pair of media and a direction, not to a material

Snell’s law is one line and the interesting content is entirely in which index is larger. n₁ sin i = n₂ sin r, with both angles from the normal. If the second medium is denser, sin r is smaller than sin i, the ray bends towards the normal, and the refracted angle is always less than the incident one — so there is no angle at which the refracted ray can escape along the surface and no critical angle exists in that direction at all. If the second medium is rarer, the ray bends away, the refracted angle outruns the incident one, and sooner or later it reaches 90°. That is the critical angle, and past it there is no refracted ray to speak of.

So the critical angle is a property of a pair of media AND a direction. This is where most of the confusion on the topic lives. People say “the critical angle of glass is 41.14°”, which is shorthand for “from crown glass into air”. From crown glass into WATER the same glass has a critical angle of 61.0°, because the ratio is 1.33/1.52 rather than 1/1.52. From air into crown glass there is no critical angle whatsoever. If you remember the direction, the formula arcsin(n₂/n₁) looks after itself: it only has a solution when n₂/n₁ is less than 1, which is exactly the condition for the second medium to be the rarer one. When the argument exceeds 1, this page prints nothing rather than a number, because nothing is the answer.

Where this goes wrong, before anything else: radians. The arcsine in every calculator, every spreadsheet and every programming language returns radians. For the default inputs on this page it returns 0.863 313, and the answer is 49.4642°. The two differ by a factor of 57.295 8, and a page that printed the first with a degree sign after it would be wrong in a way that no reader could detect without doing the problem themselves. The same trap waits going in: sin(30) in radians is −0.988, not 0.5, and Snell’s law fed that returns a negative sine and no answer. This page converts in both directions and prints the radian value as its own row so the two are visibly different quantities. If you are building this calculation yourself, that conversion is the first thing to test.

The second thing to get right is the normal. Both angles in Snell’s law are measured from the perpendicular to the surface, not from the surface. A ray described as “at 30° to the glass” is at 60° of incidence, and substituting 30 gives an answer that is wrong by a mile rather than wrong by a sign. The giveaway is the extreme cases: at normal incidence, zero degrees from the normal, nothing bends at all whatever the two indices are, which is why a window does not distort the view straight through it; at grazing incidence, nearly 90°, almost everything reflects, which is why the road ahead of you looks like a mirror when wet and the road at your feet does not.

Two things that look like the critical angle and are not. The Brewster angle, arctan(n₂/n₁), is where the REFLECTED light becomes completely polarised; it is 56.66° for air into crown glass, it exists in both directions because arctan accepts any positive argument, and it has nothing to do with the refracted ray disappearing. And total internal reflection does not switch on at the critical angle like a light: the reflectance climbs steeply over the last several degrees before it, so the transmitted ray is already faint and smeared along the surface well short of the critical angle. The critical angle is where the transmitted ray’s ANGLE reaches 90°, not where its brightness reaches zero, and treating the two as the same thing is a subtle and common slip.

Frequently asked questions

Why is there sometimes no critical angle?

Because a critical angle only exists going from a denser medium into a rarer one. Going the other way the ray bends TOWARDS the normal, so the refracted angle is always smaller than the incident one and can never reach 90° — even at a grazing 90° incidence it only reaches arcsin(n₁/n₂). Algebraically, arcsin(n₂/n₁) has no solution when n₂/n₁ exceeds 1. So the critical-angle row on this page is deliberately blank whenever the second index is the larger one, and that blank is the answer.

Is the critical angle a property of the material?

No, and this is the most useful correction on the page. It belongs to a PAIR of media and a DIRECTION. Crown glass to air is 41.14°; crown glass to water is 61.0°; air to crown glass does not exist. When somebody says “the critical angle of glass” they mean glass to air, and the phrase hides the second medium. Diamond’s famously small 24.41° is likewise diamond to air: in water it would be 33.3°, which is part of why a diamond looks dull underwater.

My calculator gives 0.863 and the answer is 49.5. What went wrong?

Nothing — 0.863 is the same angle in radians. Arcsine returns radians in every calculator in radian mode, in every spreadsheet, and in every programming language, and 0.863 313 radians is 49.4642 degrees. Multiply by 180/π, which is 57.295 8. The same conversion is needed on the way IN: sin(30) in radians is −0.988, not 0.5. This page prints both the degree and the radian value so the two cannot be mistaken for each other.

Which angle is measured from the surface and which from the normal?

Both angles in Snell’s law are from the NORMAL, the perpendicular to the surface. Nothing in optics is conventionally measured from the surface, so if a question describes a ray as being at some angle “to the glass” or “to the surface”, take the complement before substituting: 30° to the surface is 60° of incidence. The sanity check is that zero degrees should mean no bending at all, and zero from the normal does while zero from the surface does not.

Is the Brewster angle the same as the critical angle?

No, they are different phenomena that happen to be angles. The critical angle is where the REFRACTED ray reaches 90° and beyond which it ceases to exist; it is arcsin(n₂/n₁) and only exists going into a rarer medium. The Brewster angle is where the REFLECTED ray becomes completely polarised; it is arctan(n₂/n₁), it exists in both directions, and the refracted ray is entirely unaffected. For air into crown glass they are 56.66° and nothing at all respectively; for crown glass into air they are 33.34° and 41.14°.

Does total internal reflection start abruptly at the critical angle?

The refracted ray does — past the critical angle there is none, exactly. The brightness does not. The fraction of light reflected rises steeply over the last several degrees before the critical angle, so well short of it the transmitted beam is already dim and smeared along the surface, and almost all the energy is in the reflection. The critical angle marks where the refracted ray’s ANGLE reaches 90°, not where its intensity reaches zero, and this page computes angles only — it says nothing about how much light goes each way.

Related calculators

References

NCERT, Physics Textbook for Class XII, Part II, Chapter 9, Ray Optics and Optical Instruments (read 11 November 2026). Section 9.2.1 states the Cartesian sign convention: all distances measured from the pole of the mirror or the optical centre of the lens, distances measured in the same direction as the incident light taken as positive, heights above the axis positive. Equation 9.7 is the mirror equation 1/v + 1/u = 1/f and Equation 9.9 gives m = −v/u; Equation 9.23 is the thin lens formula 1/v − 1/u = 1/f and Equation 9.24 gives m = v/u. Table 9.1 lists refractive indices against air with their critical angles: water 1.33 and 48.75°, crown glass 1.52 and 41.14°, diamond 2.42 and 24.41°.
A caution about Table 9.1 that this batch could not resolve. The copy read on 11 November 2026 gives dense flint glass as 1.62 with a critical angle of 37.31°. Those two numbers are not consistent: arcsin(1/1.62) is 38.12°, and 37.31° requires an index of 1.650. Either the index in the printed table is 1.65 and the copy read here was garbled, or the row itself is internally inconsistent. This batch could not establish which, so no dense flint glass figure is printed anywhere in these eight records. The three rows that are used — water, crown glass and diamond — were each checked the same way and each reproduces its own stated critical angle to better than 0.01°.
The International System of Units (SI), 9th edition, Bureau International des Poids et Mesures, 2019. The metre is defined by fixing the speed of light in vacuum at c = 299 792 458 m/s exactly, so c carries no uncertainty at all: it is a definition, not a measurement, and every refractive index on these pages divides that exact number. The same brochure fixes the Boltzmann and Avogadro constants exactly, which makes the molar gas constant R = 8.314 462 618 153 24 J mol⁻¹ K⁻¹ exact as well — and that is what the speed-of-sound figures here are derived from rather than copied out of a table.
A worked demonstration of why the sources on this subject are cited narrowly. A widely circulated set of Class XII ray-optics notes, read 11 November 2026 alongside the NCERT chapter, gives crown glass as “about 1.5” with a critical angle of “about 41.8°”. Those two agree with each other — arcsin(1/1.5) really is 41.81° — and disagree with NCERT’s 1.52 and 41.14°. Neither is wrong about physics; they are different glasses, or the same glass rounded differently, and a student who substitutes one source’s index into another source’s answer is out by two thirds of a degree with nothing on the page to tell them. Every index on these pages comes from the NCERT table and says so.
A correction to this project’s own brief, recorded here because the next batch will need it. Both the physics brief and the authoring module’s docstring list deg and rad among the engine’s functions. Neither exists. Probed directly against assets/ce.js, both raise “unknown function”, and the test suite’s own list of permitted identifiers omits them too. Every angle on this page is therefore converted by hand — multiply by π/180 going in, by 180/π coming out — and pi is available as a constant. The brief’s own instruction is that where it and the code disagree, the code is right.