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Radians
(KS3, Year 7)
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.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 = 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.

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

^{3π} ⁄ _{2} is three quarters of a rotation.

2π is a full angle, a whole rotation.
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