What Are Radians?

What Are Radians?

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


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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.

See Also

What is an angle? What is pi?