Radian measure: the radian angle, converting between degrees and radians and the arc length ℓ = θ × r
One radian is the angle at the centre of a circle that cuts off an arc as long as the radius. A full turn has an arc of one whole circumference 2πr, so a full turn is 2π radians. Hence π radians = 180°. To change degrees to radians multiply by π/180; to change radians to degrees multiply by 180/π. One radian is about 57.3°. When a radian measure has no unit written it is understood to be in radians. The arc cut off by a central angle θ (in radians) in a circle of radius r has length ℓ = θ × r. It works only when θ is in radians; for degrees use θ × π/180 first.
| Degrees | 0° | 30° | 45° | 60° | 90° | 180° | 270° | 360° |
|---|---|---|---|---|---|---|---|---|
| Radians | 0 | π/6 | π/4 | π/3 | π/2 | π | 3π/2 | 2π |
String and pencil. Tie a pencil to a 20 cm string, hold the other end and draw an arc of length exactly 20 cm along the paper’s edge. Mark the angle: that is 1 radian. Measure it with a protractor and compare with 57.3°.