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Critical angle calculator

A-level Waves

The smallest angle of incidence, inside a dense material, at which total internal reflection happens.

sin(C) = 1 / n

This free calculator is for A-level physics. Choose the quantity you want to find, enter the other values with their units, and it shows the working in the layout used in exams, with the answer to the number of significant figures you choose. The values shown are an example. Change them to your own.

Quantity to find

41.8°

Working

  1. Write the equation
    sin(C) = 1 / n
  2. Rearrange to make C the subject
    C = sin−1(1 / n)
  3. Substitute the values
    C = sin−1(1 ÷ 1.5)
  4. Calculate
    C = 41.8103°
  5. Give the answer to 3 significant figures
    C = 41.8°

Worked example

Find C, the critical angle. Given: refractive index of the material (the other side is air) n = 1.50.

  1. Write the equation
    sin(C) = 1 / n
  2. Rearrange to make C the subject
    C = sin−1(1 / n)
  3. Substitute the values
    C = sin−1(1 ÷ 1.5)
  4. Calculate
    C = 41.8103°
  5. Give the answer to 3 significant figures
    C = 41.8°

Symbols and units

SymbolQuantitySI unit
CCritical angle°
nRefractive index of the material (the other side is air)no unit

Critical angle: rearranged

  • To find critical angle: C = sin−1(1 / n)
  • To find refractive index of the material (the other side is air): n = 1 / sin(C)

Exam tip

Total internal reflection needs light to travel from a more dense to a less dense material, and the angle of incidence to exceed C.

Common questions

What is the formula for critical angle?

The equation is sin(C) = 1 / n, where C is the critical angle (°); n is the refractive index of the material (the other side is air).

What units should I use for critical angle?

In SI units: critical angle in °, refractive index of the material (the other side is air) in no unit (a ratio). The calculator also accepts common alternatives such as kilometres, grams or minutes, and converts them for you.

How do I rearrange sin(C) = 1 / n?

Rearranging gives these forms: to find critical angle: C = sin⁻¹(1 / n). to find refractive index of the material (the other side is air): n = 1 / sin(C).

Practise critical angle questions

Press the button for a question with random numbers. Work it out on paper first, then type your answer.

Related

Read the full explanation on the Critical angle formula page. More waves equations: Wave speed, Frequency and time period, Magnification, Refractive index, Snell’s law, Double-slit fringe spacing, Diffraction grating, Intensity. See the full list on the a-level physics equation calculator page.