▸▸▸

Radians

(KS3, Year 7)

Radians are a unit of measurement of an angle. There are 2π radians in a full rotation.

Dictionary Definition

The Oxford English Dictionary defines a degree as "a unit of angle, equal to the angle subtended at the centre of a circle by an arc equal in length to the radius."

Definition of a Radian

1 radian is the angle found when the radius is wrapped around the circle.

More generally, the angle in radians is equal to:

the arc length divided by the radius.
arc length of a circle with radius 1 (the unit circle).

A Formula to Define a Radian

We have seen that the angle in radians is equal to the arc length divide by the radius. We can use this to define a formula. The angle in radians is found using the formula:

angle in radians equals arc length divided by radius

angle in radians equals arc length divided by radius

In this formula, θ is the angle in radians, s is the arc length and r is the radius. The image below shows what we mean by angle, arc length and radius:

radians image showing angle, arc length and radius

radians image showing angle, arc length and radius

Lesson Slides

The slider below shows more about the definition of radians.

2π Radians in a Circle

There are 2π radians in a full rotation (a circle).

2 pi radians in a circle

Using the definition of radians, can you work out why there are 2π radians in a circle? See the Note to find out why.

Why Are There 2π Radians in a Circle?

The angle in radians is the arc length divided by the radius. For a full circle, the arc length is the circumference. The circumference of a circle with a radius of r is 2πr.

Angle = Arc length ⁄ Radius

Angle = Circumference ⁄ Radius

Angle = 2πr ⁄ r

Angle = 2π

This is why the full angle in radians is 2π radians.