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What Are Radians? (Mathematics Lesson)

What Are Radians?

Radians are a unit of measurement of an angle.

There are 2π radians in a full rotation.

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).
The slider below shows more about the definition of radians:

Important Angles in Radians

  • 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.

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Note
WHAT IS AN ANGLE?

An angle is a measure of the rotation between two straight lines that share a common endpoint.

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.



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

IMPORTANT ANGLES IN RADIANS

The important angles in radians can be found by proportion with a whole circle.

A right angle is a quarter of a full angle.
¼ × 2π = π2 radians.

A straight angle is a half of a full angle.
½ × 2π = π radians.

Three quarters of a rotation is three quarters of a full angle.
¾ × 2π = 2 radians.

OTHER MEASURES OF ANGLES

The degree is another measure of angles.

There are 360 degrees in one rotation.

Degrees are denoted by the symbol °.

Read more about degrees