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First exams 2025

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Radians & Angular Displacement (CIE A Level Physics)

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Radians & angular displacement

Angles in radians

  • A radian (rad) is defined as:

The angle subtended at the centre of a circle by an arc equal in length to the radius of the circle

Visual definition of radian

6-1-1-one-radian_sl-physics-rn

When the angle is equal to one radian, the length of the arc (S) is equal to the radius (r) of the circle

  • Radians are commonly written in terms of π
  • The angle in radians for a complete circle (360°) is equal to:

fraction numerator circumference space of space circle over denominator radius end fraction space equals fraction numerator space 2 straight pi r over denominator r end fraction space equals space 2 straight pi


  • Use the following equation to convert from degrees to radians:

theta degree space cross times space open parentheses straight pi over 180 close parentheses space equals space theta space rad

Table of common degrees to radians conversions

Degrees (°) Radians (rads)
360 2 straight pi
270 fraction numerator 3 straight pi over denominator 2 end fraction
180 straight pi
90 straight pi over 2

Angular displacement

  • In circular motion, it is more convenient to measure angular displacement in units of radians rather than units of degrees
  • Angular displacement is defined as:

The change in angle, in radians, of a body as it rotates around a circle

  • This can be summarised in equation form:

increment theta space equals space fraction numerator distance space travelled space around space the space circle over denominator radius space of space the space circle end fraction

  • Where:
    • Δθ = angular displacement, or angle of rotation (radians)
    • S = length of the arc, or the distance travelled around the circle (m)
    • r = radius of the circle (m)

  • Note: both distances must be measured in the same units e.g. metres

Visual representation of angular displacement equation

6-1-1-angle-in-radians_sl-physics-rn

An angle in radians, subtended at the centre of a circle, is the arc length divided by the radius of the circle

Worked example

Convert the following angular displacement into degrees:WE - Radians conversion question image, downloadable AS & A Level Physics revision notes

Answer: 

Step 1: Rearrange the degrees to radians conversion equation

degrees space rightwards arrow space radians space rightwards double arrow space theta degree space cross times space straight pi over 180 space equals space theta space rad

radians space rightwards arrow space degrees space rightwards double arrow space theta space rad space cross times fraction numerator space 180 over denominator straight pi end fraction space equals space theta degree

Step 2: Substitute the values to calculate

straight pi over 3 rad space cross times fraction numerator space 180 over denominator straight pi end fraction space equals space fraction numerator 180 degree over denominator 3 end fraction space equals space 60 degree

Examiner Tip

  • You will notice your calculator has a degree (Deg) and radians (Rad) mode
  • This is shown by the “D” or “R” highlighted at the top of the screen
  • Remember to make sure it’s in the right mode when using trigonometric functions (sin, cos, tan) depending on whether the answer is required in degrees or radians
  • It is extremely common for students to get the wrong answer (and lose marks) because their calculator is in the wrong mode - make sure this doesn’t happen to you!

 Radians on calculator, downloadable AS & A Level Physics revision notes

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Leander

Author: Leander

Expertise: Physics

Leander graduated with First-class honours in Science and Education from Sheffield Hallam University. She won the prestigious Lord Robert Winston Solomon Lipson Prize in recognition of her dedication to science and teaching excellence. After teaching and tutoring both science and maths students, Leander now brings this passion for helping young people reach their potential to her work at SME.