Arc Length and Area of a Sector: GCSE Maths Guide
Quick answer: A sector is a "pizza slice" of a circle. Find what fraction of the circle it is, θ ÷ 360, then take that fraction of the whole circle. Arc length = (θ ÷ 360) × 2πr. Sector area = (θ ÷ 360) × πr². The perimeter of a sector is the arc length plus two radii.
The two formulas
Arc length: (θ ÷ 360) × 2πr
Area of a sector: (θ ÷ 360) × πr²
Here θ is the angle at the centre in degrees, and r is the radius. You don't need to memorise anything new: it's the circumference or area of the full circle, scaled down.
Worked example 1: arc length and area
A sector has radius 6 cm and angle 60°. Find the arc length and the area.
60 ÷ 360 = 1/6 of the circle.
Arc length = 1/6 × 2π × 6 = 2π ≈ 6.28 cm.
Area = 1/6 × π × 36 = 6π ≈ 18.85 cm².
Worked example 2: perimeter of a sector
Find the perimeter of the same sector.
Perimeter = arc + two radii = 2π + 6 + 6 ≈ 18.28 cm.
A lot of students forget the two straight edges. Check the question: "perimeter" means all the way round.
Worked example 3: finding the angle
A sector has area 50 cm² and radius 8 cm. Find the angle, to 1 decimal place.
50 = (θ ÷ 360) × π × 64
θ = 50 × 360 ÷ (64π) ≈ 89.5°.
Worked example 4: area of a segment
A sector has radius 10 cm and angle 90°. Find the area of the segment between the chord and the arc.
Sector: (90 ÷ 360) × π × 100 = 25π ≈ 78.54 cm².
Triangle: ½ × 10 × 10 × sin 90° = 50 cm².
Segment = 78.54 − 50 ≈ 28.5 cm².
This is the same "whole minus the bit you don't want" method used for any shaded area. See our area of the shaded region guide. The triangle uses ½ab sin C, covered in our sine rule vs cosine rule guide.
Exam tips
Write the fraction first. θ ÷ 360 makes the rest clear.
Leave answers in terms of π if the question asks.
Use the radius, not the diameter.
Keep full calculator values until the last line.
Tips are MMS editorial guidance.
Common mistakes
Using 2πr for area or πr² for arc length. Arc is a length; area is squared units.
Leaving out the radii in a perimeter question.
Rounding the sector before subtracting in a segment question.
When extra support might help
Support is worth considering when circle questions keep losing marks, or when a student wants exam-style practice before mocks.
A tutor can't guarantee a grade, but can build method and confidence. Math Make Smart offers online maths tutoring in the UK, and you can book a free trial lesson.
Frequently asked questions
What is the formula for the area of a sector?
(θ ÷ 360) × πr².
What is the formula for arc length?
(θ ÷ 360) × 2πr.
How do you find the perimeter of a sector?
Add the arc length and two radii.
What is a segment?
The area between a chord and the arc. Find it as sector minus triangle.