# 6.7 Radians & Angular Displacement

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

• Radians are used whenever describing the angular displacement of objects in circular motion

• Angular displacement can be calculated using the equation:

• 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)

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

• If an angle of 360o = 2π radians, then 1 radian in degrees is equal to:

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

Table of common degrees to radians conversions

#### Worked example

Convert the following angular displacement into degrees:

#### Exam Tip

• 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 when using trigonometric functions - make sure this doesn’t happen to you!

## Angular Displacement

• The angular displacement Δθ is the ratio of:

• Angular displacement describes the change in angle, in radians, of a body as it moves in a circle
• This angle is measured with respect to the centre of orbit

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

#### Exam Tip

Since the equation for angular displacement gives the angle in radians, make sure you're comfortable with then converting to degrees if you need to for the question!

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