## The Lesson

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."

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:

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:

## 2π Radians in a Circle

There are 2π radians in a full rotation (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 = 2πr ⁄ r

Angle = 2π

• π2 is a right angle, a quarter of a rotation.

• π is a straight angle, a half of a rotation.

• 2 is three quarters of a rotation.

• 2π is a full angle, a whole rotation.