What is a radian?
In circular measure we measure angles using radians rather than degrees. One radian is the angle subtended at the centre of a circle by an arc whose length is equal to the radius of the circle. Since the circumference of a circle is 2πr, a complete revolution contains exactly 2π radians. This gives the fundamental relationship 2π rad = 360°, and therefore π rad = 180°.
Key formula
π rad = 180° · degrees → radians: multiply by π/180 · radians → degrees: multiply by 180/π
Converting between units
To change degrees into radians, multiply by π/180. To change radians into degrees, multiply by 180/π. Keep answers in terms of π when the numbers are exact; otherwise round to a suitable number of decimal places, using π ≈ 3.142.
Worked example
Convert 60° to radians: 60 × π/180 = π/3 ≈ 1.047 rad. Convert 1.5 rad to degrees: 1.5 × 180/π = 270/π ≈ 85.94°. Convert 120° to radians: 120 × π/180 = 2π/3 rad.
Why radians matter
Radians make the formulas for arc length (s = rθ) and area of a sector (A = ½r2θ) simple, but only when the angle θ is measured in radians. Using degrees directly in these formulas gives wrong answers. Common conversions worth memorising are 30° = π/6, 45° = π/4, 90° = π/2 and 180° = π.
Remember
- π rad = 180° is the key bridge between the two units.
- Set your calculator to RAD mode when working in radians.
- A full turn is 2π rad, a half turn is π rad, and a quarter turn is π/2 rad.