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."

Definition of a Radian

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

1 radian 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 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

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.

Important Angles in Radians

  • π ⁄ 2 is a right angle, a quarter of a rotation.

    pi by 2 radians is a quarter turn
  • π is a straight angle, a half of a rotation.

    pi radians is a half turn
  • 3π ⁄ 2 is three quarters of a rotation.

    3 pi by 2 radians is three quarters of a turn
  • 2π is a full angle, a whole rotation.

    2 pi radians is a whole turn