Radians

What it is

A radian is a measure of angle that starts from the circle instead of from the number 360. Draw a circle of radius r and mark an arc along the edge that is also r long. The angle between the radii to the ends of that arc is 1 radian.

The circumference is 2πr, so the radius fits 2π ≈ 6.283 times around the circle. A full turn is therefore 2π radians, a half turn π and a quarter turn π/2:

• 180° = π rad ≈ 3.14 rad
• 90° = π/2 rad ≈ 1.57 rad
• 60° = π/3 rad, 45° = π/4 rad, 30° = π/6 rad
• 1 rad = 180°/π ≈ 57.3°

With radians the arc length is simple: s = r · θ. An angle of 2 rad in a circle of radius 5 cm gives an arc of 10 cm, see arc length in radians. The sector area is r² · θ / 2, here 25 · 2 / 2 = 25 cm² (sector area in radians).

What is not a radian:

• 90 is not π/2 rad. 90 rad is just over 14 turns: 90 / (2π) ≈ 14.3.
• “π” as a degree count. π ≈ 3.14 is a number; π rad is 180°, but π° is only about 0.05 rad.
• A length in centimetres. A radian is the ratio of arc to radius, so the units cancel (cm / cm). That is why it is often called dimensionless.

Why it matters

Radians are the measure in which formulas come out simple. Arc length s = r · θ and sector area r²θ/2 have no extra constants. In calculus the derivative of sin x is cos x only when x is in radians; in degrees a factor π/180 appears.

To change units there are degrees to radians and radians to degrees. On the unit circle angles are often given in radians.

Common misconceptions

• “Radians are a different kind of angle.” It is the same angle in another unit, like metres and feet.
• “An angle in radians has to contain π.” π/3 is convenient to write because 60° is a simple part of a turn, but 1, 2 and 0.5 rad are just as valid. 1 rad is about 57.3°.
• “The calculator gives the right answer in any mode.” sin 90 is 1 in degree mode but about 0.894 in radian mode, because 90 rad is 2.04 rad after 14 whole turns. Check the mode.
• “π is 180.” π rad corresponds to 180°; the number π is about 3.14.

Common questions