(KS3, Year 7)
The LessonRadians are a unit of measurement of an angle. There are 2π radians in a full rotation.
Dictionary DefinitionThe 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 Radian1 radian is the angle found when the radius is wrapped around the circle.
More generally, the angle in radians is equal to:
A Formula to Define a RadianWe 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 SlidesThe slider below shows more about the definition of radians. Open the slider in a new tab
2π Radians in a CircleThere 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 = 2πThis 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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