To find an arc length, use s = rθ, where r is the radius and θ is the angle at the centre in radians. This is the shortest formula in radians, arcs and sectors, and the place where most unit errors begin.
Where does s = rθ come from?
A full circle has circumference 2πr and angle 2π. The arc for an angle θ is the fraction θ/2π of the whole circumference: (θ/2π) × 2πr. The 2π cancels, leaving s = rθ.
So the formula says: arc length = radius × angle in radians. Doubling the angle doubles the arc, which matches what you would expect.
How do you use it, step by step?
- Write the angle in radians. If it arrives in degrees, use the conversion from the previous lesson.
- Write the formula s = rθ and substitute r and θ.
- Work out the value and keep π in the answer if the question wants an exact value.
- Attach the unit of the radius.
- Rearrange if needed: r = s/θ for a radius, θ = s/r for an angle.
Worked example
A sector has radius 9 cm and angle 140°. Find the arc length to 3 significant figures.
Step 1, convert: 140 × π/180 = 7π/9 rad.
Step 2, substitute: s = rθ = 9 × 7π/9.
Step 3, simplify: the 9s cancel, so s = 7π.
Step 4, evaluate: 7π = 21.991… so s = 22.0 cm to 3 significant figures.
Check with fractions of a circle: 140/360 × 2π × 9 = 140/360 × 56.549 = 21.991. ✓ Both routes agree.
The mistake to watch for
A common slip is to put the degree angle straight into s = rθ.
Mistaken working: s = 9 × 140 = 1260 cm
The formula needs radians. An arc of 1260 cm on a circle of radius 9 cm is also impossible, since the whole circumference is only about 56.5 cm.
The correction is to convert first, and to do a size check: an arc can never be longer than the full circumference 2πr. That check takes five seconds and catches the error.
Check yourself
1. A sector has radius 5 cm and angle 0.6 rad. Find the arc length.
Show answer
s = rθ = 5 × 0.6 = 3.
3 cm
2. A sector has radius 12 cm and angle 5π/6 rad. Find the arc length in terms of π and to 3 significant figures.
Show answer
s = 12 × 5π/6 = 10π = 31.416…
10π cm, which is 31.4 cm to 3 significant figures.
3. An arc of length 14.4 cm subtends an angle of 1.8 rad at the centre. Find the radius.
Show answer
r = s/θ = 14.4 ÷ 1.8 = 8.
8 cm. Check: 8 × 1.8 = 14.4. ✓
Where this leads next
Once arc length feels automatic, move on to calculating a sector area, which uses the same angle in radians. The non-calculator working trainer is useful for keeping answers exact with π.
If the formula is clear but you still choose the wrong angle under pressure, our teachers can work on that through online one-to-one Additional Mathematics tuition.