5.2PureCore

Arc Length

When the angle is measured in radians, the length of a circular arc is simply the radius times the angle. This clean formula is the main reason radians are preferred over degrees.

25 min Video by Zeeshan Zamurred Radians

Edexcel A Level Maths: 5.2 Arc LengthWatch the full walkthrough before the notes below.

Open on YouTube

What you'll be able to do

Use s = rθ for arc length
Work with θ in radians
Find perimeters of sectors
Solve for an unknown radius or angle

The arc length formula

For a sector of radius $r$ with angle $θ$ in $radians$ , the arc length is $s = r θ$ . The angle must be in radians — convert first if given in degrees.

s = r θ (θ in radians)

Arc length = radius × angle.

Perimeter of a sector

The perimeter of a sector is the arc plus the two radii: $P = r θ + 2 r$ .

s = 5 \times 0.8 = 4

cm.

P = 4 + 2 (5) = 14

cm.

AnswerArc

4

cm, perimeter

14

cm.

Tip — Perimeter includes both straight radii — don’t forget the +2r.

Formula recap

s = r θ

Arc length (θ in radians).

P = r θ + 2 r

Sector perimeter.

Common mistakes to avoid

Using s = rθ with θ in degrees.

θ must be in radians for s = rθ.

Forgetting the two radii in a sector perimeter.

Perimeter = arc + 2r.

Key takeaways

s = rθ with θ in radians.
Convert degrees to radians first.
Sector perimeter = rθ + 2r.

Test yourself

Ready to lock in Arc Length? Pick a mode and earn XP & Dobloons.