The arc length formula
An arc is part of the circumference of a circle. When the angle θ at the centre is measured in radians, the length of the arc is directly proportional to the angle. This gives the simple formula s = rθ, where s is the arc length, r is the radius and θ is the angle in radians. The formula works only when θ is in radians, so always convert from degrees first.
Key formula
s = rθ (θ in radians) · rearranged: r = s/θ and θ = s/r
Rearranging the formula
The same equation can be rearranged to find any of the three quantities. If you know the arc length and radius, the angle is θ = s/r. If you know the arc length and angle, the radius is r = s/θ. Always keep the units consistent, giving lengths in the same unit and angles in radians.
Worked example
A circle has radius 5 cm and an arc subtends an angle of 1.2 rad at the centre. The arc length is s = rθ = 5 × 1.2 = 6 cm. If instead an arc of length 12 cm is drawn on a circle of radius 6 cm, the angle is θ = s/r = 12/6 = 2 rad.
Perimeter of a sector
A sector is bounded by an arc and two radii. Its perimeter is the arc length plus the two straight radii, that is P = rθ + 2r. For a circle of radius 5 cm with θ = 1.2 rad, the perimeter is 6 + 10 = 16 cm.
Remember
- θ must be in radians before using s = rθ.
- Convert degrees using θ = degrees × π/180.
- Perimeter of a sector = arc length + 2r.