PhysicsICSE Class 10 · Total Internal Reflection

Total Internal Reflection in Prisms: Turning Light Through 90° and 180°, and Inner-Face Geometry (ICSE Class 10)

Updated 15 October 2026 · Fact-checked

A right-angled isosceles (45°–45°–90°) glass prism totally reflects a ray that meets a face inside at 45°, because glass's critical angle (about 42°) is less than 45°. Entering normally through a short face, the ray turns through 90°; entering normally through the hypotenuse, it is reflected twice and turns through 180°.

Understand Total Internal Reflection in Prisms

A total reflecting prism uses TIR at an inner face in place of a mirror. For this the ray must meet that face from inside at an angle greater than the critical angle of the prism material (about $42°$ for glass).

Finding the angle at an inner face. A ray entering a face normally goes straight on. Then:

In a prism of apex angle $A$ , a ray entering one face normally meets the other face of that apex at an angle of incidence equal to $A$ .
In a right-angled prism ABC (right angle at A), a ray entering normally through AB travels parallel to AC and meets the hypotenuse BC at an angle of incidence equal to $∠ B$ .

Always draw the normal at the new face and measure from it.

Deviation by one total reflection. If a ray is totally reflected at angle of incidence $i$ , it is turned through

$δ = 180° - 2 i .$

1. Turning through 90° (45°–45°–90° prism). The ray enters normally through one of the short faces, goes straight, meets the hypotenuse at $45° > 42°$ , is totally reflected ( $δ = 180° - 90° = 90°$ ), and leaves normally through the other short face. Used in periscopes.

2. Turning through 180° (same prism, hypotenuse facing the light). The ray enters normally through the hypotenuse, meets one short face at $45°$ (TIR), crosses to the other short face, meets it at $45°$ (TIR again) and leaves through the hypotenuse travelling back parallel to its original path. The two reflections ( $90° + 90°$ ) turn it through $180°$ . The top and bottom of a beam swap places, so the image is inverted: such an erecting prism is used in binoculars. Bicycle and road reflectors use many tiny prisms of this kind to send headlight beams straight back.

3. Equilateral prism. A ray entering one face normally meets the second face at $60° > 42°$ : TIR, deviation $180° - 120° = 60°$ . It then meets the third face normally and leaves undeviated, so the total deviation is $60°$ .

When TIR fails. A 45° prism works only if $C < 45°$ , i.e. $μ > \frac{1}{s i n 45°} = 2 \approx 1.414$ . Glass (1.5) works; water (1.33) or a plastic of μ 1.4 does not: the ray would partly refract out of the hypotenuse.

Why prisms rather than plane mirrors? (qualitative)

TIR reflects practically all the light; a mirror absorbs some, so images are brighter.
A prism gives one sharp image; a thick glass mirror gives multiple faint images by partial reflections.
There is no silvering to tarnish or peel, so prisms last longer.

Total Internal Reflection in Prisms: formulas and key facts

Deviation by one total reflection: $δ = 180° - 2 i$; i = angle of incidence at the reflecting face.
Inner angle for normal entry: $i_{second face} = A$; A = angle between the entry face and the second face.
Condition for a 45° prism to work: $C < 45° ⟺ μ > 2 \approx 1.414$; Glass (μ 1.5, C ≈ 42°) qualifies.

How to solve Total Internal Reflection in Prisms questions

Trace the ray face by face: at each face decide 'straight through', 'refract' or 'TIR'.

1At the entry face: if the ray is normal, it goes straight; otherwise use Snell's law.
2Find the angle of incidence at the next face by geometry (apex angle, or ∠B for a right-angled prism), measured from the normal.
3Compare with C: if $i > C$ , TIR with angle of reflection $= i$ ; if $i < C$ , refraction out, bending away from the normal.
4Repeat at the next face until the ray leaves.
5Total deviation = angle between the original and final directions; for each TIR, $180° - 2 i$ .
6Draw arrows on every ray and mark angles from the normals.

Total Internal Reflection in Prisms in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

A bicycle reflector

Suppose a car's headlights shine on the rear reflector of a bicycle at night. The reflector's back surface is moulded into many tiny 45° prism shapes.

Why does the light go straight back towards the driver?

Try it first, then show the solutionSolution

Light enters through the flat front and meets two inclined back faces in turn at about $45°$ , more than the critical angle.
It is totally reflected twice and turned through $180°$ , travelling back towards the headlights.

Answer: Each tiny prism returns the light by two total internal reflections.

Memory hook: Two TIRs make a U-turn.

Binoculars

Imagine looking through a pair of prism binoculars. A simple two-lens telescope would show the scene upside down.

How do the prisms inside give an upright, bright view?

Try it first, then show the solutionSolution

Pairs of 45°–45°–90° prisms turn the light through $180°$ each by two total reflections, folding the long light path into a short case.
Each such prism inverts the image; arranged at right angles, they turn it upright, and TIR loses almost no light.

Answer: Erecting prisms fold the path, invert the image back upright and keep it bright.

Memory hook: Fold, flip, and stay bright.

Tricks to remember Total Internal Reflection in Prisms

Short memory aids. Say each one aloud once, then use it on a question the same day.

Short face → 90, long face → 180
Light in through a Short face turns 90°; in through the Long face (hypotenuse) turns 180°.
Use it when: Choosing the prism orientation.
180 minus twice i
Each total reflection turns the ray by 180° − 2i.
Use it when: Deviation at a reflecting face.
Normal in, apex angle out
Enter normally, and the angle at the next face equals the apex angle between the two faces.
Use it when: Inner-face geometry.
Root two rule
A 45° prism needs μ above √2 (1.414); glass at 1.5 passes, water fails.
Use it when: Questions on prism materials.

Total Internal Reflection in Prisms: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Material for a 45° prism · 1 mark · MCQBoard level

A right-angled isosceles prism is to turn a ray through $90°$ by total internal reflection at its hypotenuse. The critical angle of its material must be:
(a) less than $45°$
(b) exactly $90°$
(c) more than $45°$
(d) more than $60°$

Try it first, then show the solutionSolution

The ray meets the hypotenuse at $45°$ ; TIR needs $45° > C$ .

Answer: (a) less than $45°$

Example 2: Choosing the face of entry · 3 marks · Short answerBoard level

For a 45°–45°–90° glass prism, state (a) through which face light must enter normally to be turned through $90°$ , and the instrument that uses this; (b) through which face it must enter to be turned through $180°$ , the number of total reflections, and one use.

Try it first, then show the solutionSolution

(a) One of the two shorter faces; one TIR at the hypotenuse; used in a periscope. (1½)
(b) The hypotenuse face; two TIRs, one at each short face; used in binoculars (or bicycle reflectors). (1½)

Answer: (a) A short face; periscope (b) The hypotenuse; two; binoculars

Example 3: When a 45° prism fails · 3 marks · Short answerHarder board level

A 45°–45°–90° prism is made of a material of refractive index 1.3. A ray enters one short face normally. (a) Find the critical angle of the material. (b) Is the ray totally reflected at the hypotenuse? (c) If not, find the angle at which it leaves the hypotenuse and its deviation there.

Try it first, then show the solutionSolution

(a) $sin C = \frac{1}{1.3}$ , so $C \approx 50.3°$ . (1)
(b) No: the angle of incidence at the hypotenuse is $45° < 50.3°$ . (1)
(c) $sin r = 1.3 sin 45° \approx 0.919$ , $r \approx 66.8°$ ; deviation $= 66.8° - 45° \approx 21.8°$ . (1)

Answer: (a) About $50.3°$ (b) No (c) About $66.8°$ ; deviation about $21.8°$

Example 4: Critical angle from a grazing ray · 3 marks · Short answerHarder board level

A prism ABC has $∠ A = 90°$ , $∠ B = 50°$ and $∠ C = 40°$ . A ray of blue light enters face AB normally and is seen to emerge just grazing along face BC. (a) Find the angle of incidence of the ray at BC. (b) Hence find the critical angle and the refractive index for blue light. (c) Which colour of light, sent the same way, would be totally reflected at BC?

Try it first, then show the solutionSolution

(a) Entering AB normally, the ray travels parallel to AC and meets BC at an angle of incidence equal to $∠ B = 50°$ . (1)
(b) Grazing emergence means $i = C = 50°$ ; $μ = \frac{1}{s i n 50°} \approx 1.31$ . (1)
(c) A colour of shorter wavelength (indigo or violet), which has a larger μ and a smaller critical angle. (1)

Answer: (a) $50°$ (b) $C = 50°$ , μ ≈ 1.31 (c) Indigo or violet

Example 5: Tracing a ray across three faces · 4 marks · Long answerHOTS / competency

A glass prism ABC (μ = 1.5, C ≈ $41.8°$ ) has $∠ A = 90°$ , $∠ B = 65°$ and $∠ C = 25°$ . A ray enters face AB normally.
(a) Find its angle of incidence at BC and say what happens there.
(b) Through what angle is it turned at BC?
(c) The reflected ray next meets face AC. Find its angle of incidence there.
(d) Does it leave through AC? If so, at what angle?

Try it first, then show the solutionSolution

(a) The ray travels parallel to AC and meets BC at $i = ∠ B = 65° > 41.8°$ : TIR. (1)
(b) $δ = 180° - 2 \times 65° = 50°$ . (1)
(c) The ray was parallel to AC; turned through $50°$ towards AC, it meets AC at $50°$ to the face, so its angle of incidence is $90° - 50° = 40°$ . (1)
(d) $40° < 41.8°$ , so it refracts out: $sin e = 1.5 sin 40° \approx 0.964$ , $e \approx 74.6°$ . (1)

Answer: (a) $65°$ , TIR (b) $50°$ (c) $40°$ (d) Yes, at about $74.6°$

Example 6: Case study: choosing a prism for a periscope · 4 marks · Case-basedHOTS / competency

A school club builds a periscope with two right-angled isosceles prisms, one at each end of a tube. They have a cheap plastic of μ = 1.4 and a glass of μ = 1.5.
(a) Through what angle must each prism turn the light, and at what angle does the light meet the hypotenuse?
(b) Show which material works.
(c) What is the least refractive index that would work?
(d) Give two reasons why the glass prisms are better than two plane mirrors.

Try it first, then show the solutionSolution

(a) $90°$ each; the light meets the hypotenuse at $45°$ . (1)
(b) Plastic: $sin C = \frac{1}{1.4}$ , $C \approx 45.6° > 45°$ : no TIR, light leaks out. Glass: $C \approx 41.8° < 45°$ : TIR. Glass works. (1)
(c) $μ > \frac{1}{s i n 45°} = 2 \approx 1.414$ . (1)
(d) TIR reflects practically all the light (brighter image), and there is no silvering to tarnish; also no multiple images. (1)

Answer: (a) $90°$ ; $45°$ (b) Glass only (c) About 1.414 (d) Brighter image; no tarnishing (and a single image)

Where students lose marks in Total Internal Reflection in Prisms

The slips examiners see most often, and what to write instead.

Mistake: Not finding the angle of incidence at the inner face
TIR can only be judged after the angle at the new face is worked out by geometry, from the normal.
Wrong working
Assuming every ray inside a prism meets the next face at $45°$ , or measuring from the surface.
Right working
Normal entry: inner angle = apex angle (or $∠ B$ for a right-angled prism), measured from the normal.
Do this: Draw the normal at every face you reach.
Usually costs: Reported by CISCE examiners for 2025 Q4(i).
Mistake: Misjudging the total deviation
Deviation is the angle between the original incident direction and the final emergent direction.
Wrong working
Equilateral prism, normal entry, TIR at the second face: 'deviation = $120°$ ' or ' $30°$ '.
Right working
$δ = 180° - 2 \times 60° = 60°$ , and the ray leaves the third face undeviated: total $60°$ .
Do this: Extend the incident ray and compare it with the final ray.
Usually costs: Reported for 2025 Q1(viii).
Mistake: Assuming an isosceles prism
Prism angles must be worked out from the data, not guessed from the look of the diagram.
Wrong working
'The prism angle is $45°$ ' when only one angle is given as $90°$ and another as $52°$ .
Right working
Use the angle sum of a triangle and the given angles.
Do this: Write down all three angles before tracing the ray.
Usually costs: Reported for 2024 Q4(iii).
Mistake: No arrows on rays
Arrows show the direction of travel, which decides denser → rarer.
Wrong working
A ray path drawn as plain lines.
Right working
An arrow on every segment of the ray.
Do this: Add arrows as you draw each segment.
Usually costs: Reported for 2025 Q4(i).

Exam tips

In 2025 Q4(i), the expected path was: the ray enters normally, meets the next face at $60°$ (more than $42°$ ), is totally reflected, then meets the base at $30°$ and emerges, bending away from the normal.
In 2025 Q1(viii) (equilateral prism, normal entry, TIR at the second face), the answer was a total deviation of $60°$ .
When asked for the advantages of prisms over mirrors, give two distinct ones: brightness (no absorption) and durability (no silvering) or a single image.
Always mark the angle of incidence at the reflecting face and write 'i > C, so TIR'.

Practice questions on Total Internal Reflection in Prisms

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
When a 45°–45°–90° glass prism turns a ray through $180°$ , the number of total internal reflections inside it is:
(a) 0
(b) 1
(c) 2
(d) 3
Hint
Each reflection turns the ray through $90°$ .
Answer
(c) 2
Q2 · 2 marks · Short answerBoard level
Why are bicycle rear reflectors made with many small 45° prism shapes on the back instead of a flat mirror?
Hint
Where does the light have to go?
Answer
Each prism totally reflects light twice and turns it through $180°$ , sending it back towards the vehicle that lit it. TIR reflects almost all the light, so the reflector looks bright to the driver.
Q3 · 1 mark · Very short answerHarder board level
A ray is totally reflected inside a prism at an angle of incidence of $50°$ . Through what angle is it deviated at that face?
Hint
$δ = 180° - 2 i$ .
Answer
$180° - 100° = 80°$ .
Q4 · 3 marks · Short answerHarder board level
A glass prism ABC (C ≈ $42°$ ) has $∠ A = 90°$ , $∠ B = 55°$ , $∠ C = 35°$ . A ray enters face AB normally. (a) Find its angle of incidence at BC. (b) What happens there? (c) Through what angle is it deviated at BC?
Hint
The ray travels parallel to AC.
Answer
(a) $∠ B = 55°$ . (b) $55° > 42°$ : total internal reflection. (c) $180° - 110° = 70°$ .
Q5 · 2 marks · Short answerHarder board level
A candle is viewed in a total reflecting prism and in a thick plane mirror. Why does the prism show one bright image but the mirror several images?
Hint
Where does reflection happen in each?
Answer
In the mirror, light is partly reflected at the front glass surface and mainly at the silvered back, with further reflections inside, giving several images of different brightness. In the prism, nearly all the light is reflected at one surface by TIR, giving one bright image.
Q6 · 3 marks · Short answerHarder board level
A ray enters, at right angles, one face of a glass prism (μ = 1.5) whose apex angle is $38°$ . (a) At what angle does it strike the opposite face of the apex? (b) Is it totally reflected there? (c) If not, at what angle does it come out, and how much is it deviated?
Hint
Normal entry: the inner angle equals the apex angle.
Answer
(a) $38°$ . (b) No: $38° < 41.8°$ . (c) $sin e = 1.5 sin 38° \approx 0.923$ , $e \approx 67.4°$ ; deviation $= 67.4° - 38° \approx 29.4°$ .
Q7 · 3 marks · Short answerHOTS / competency
A ray enters one face of a glass prism (C ≈ $41.8°$ ) normally. (a) What is the smallest apex angle the prism can have if the ray is to be totally reflected at the second face? (b) For an apex angle of $50°$ , find the deviation at the second face.
Hint
The inner angle equals the apex angle.
Answer
(a) The apex angle must exceed $41.8°$ . (b) $i = 50° > 41.8°$ : TIR, $δ = 180° - 100° = 80°$ .

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Deviation of a ray entering a short face of a 45°–45°–90° glass prism normally?Show answer
$90°$ .
2.Deviation when it enters the hypotenuse normally?Show answer
$180°$ .
3.Least μ for a 45° prism to give TIR?Show answer
$2 \approx 1.414$ .
4.Deviation produced by one total reflection at $i = 30°$ ?Show answer
$120°$ .
5.Which instrument uses 180° prisms?Show answer
Binoculars.

Total Internal Reflection in Prisms: frequently asked questions

Why does a right-angled isosceles glass prism totally reflect light?

A ray entering a face normally meets the next face inside at 45°, which is greater than the critical angle of glass (about 42°). Light going from glass to air at more than the critical angle is totally reflected.

How does a prism turn light through 180°?

The ray enters normally through the hypotenuse, is totally reflected at one short face and then at the other, each time at 45°. The two 90° turns add to 180°, so it goes back parallel to its original path.

Why are prisms used instead of mirrors in periscopes and binoculars?

Total internal reflection reflects nearly all the light, giving brighter images; prisms give a single clear image; and they have no silvering to tarnish.