## 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: ## Lesson Slides

The slider below shows more about the definition of radians. Open the slider in a new tab

## 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 = Circumference ⁄ Radius Angle = 2πr ⁄ r Angle = 2π    