Arc Length and Sector Area: GCSE Maths Guide

A sector is a "pie slice" of a circle — the region between two radii and the arc between them. The arc length is the curved part of the boundary; the sector area is the area of the slice. Both are found by treating the sector angle as a fraction of the full circle (360°).

What is the difference between an arc, a sector, and a segment?

These three terms are often confused. Knowing the difference prevents errors before you even start the calculation.

Term | What it is
Arc | The curved part of the circle's circumference between two points
Sector | The region enclosed by two radii and the arc between them (a "pie slice")
Segment | The region between a chord and the arc it cuts off (NOT the pie-slice shape)
Chord | A straight line joining two points on the circumference

At GCSE, arc length and sector area are the main calculations. Segment area appears on Higher papers.

What are the formulas for arc length and sector area?

Both formulas use the idea that the sector angle θ (in degrees) is a fraction of the full circle (360°).

Arc length = (θ / 360) × 2πr = (θ / 360) × πd

Sector area = (θ / 360) × πr²

where r is the radius and θ is the angle at the centre in degrees.

These are the formulas you need to memorise. Some exam papers provide formulae sheets, but arc length and sector area are often NOT included — always check.

How do you calculate arc length?

Worked example 1: A sector has radius 9 cm and angle 80°. Find the arc length. Give your answer to 3 significant figures.

Fraction of circle = 80 / 360 = 2/9.

Circumference of full circle = 2π × 9 = 18π cm.

Arc length = (80/360) × 18π = (2/9) × 18π = 4π ≈ 12.6 cm (3 s.f.).

Worked example 2: A sector has diameter 14 cm and angle 135°. Find the arc length. Give your answer in terms of π.

Radius = 14 ÷ 2 = 7 cm.

Arc length = (135/360) × 2π × 7 = (3/8) × 14π = 42π/8 = 21π/4 cm.

How do you calculate sector area?

Worked example 3: A sector has radius 6 cm and angle 120°. Find the area. Give your answer to 1 decimal place.

Fraction of circle = 120 / 360 = 1/3.

Area of full circle = π × 6² = 36π cm².

Sector area = (1/3) × 36π = 12π ≈ 37.7 cm² (1 d.p.).

Worked example 4: A sector has radius 10 cm and angle 72°. Find the area. Give your answer in terms of π.

Sector area = (72/360) × π × 10² = (1/5) × 100π = 20π cm².

How do you find the perimeter of a sector?

The perimeter of a sector is made up of two radii plus the arc length — not just the arc alone. Forgetting to add the two radii is one of the most common mistakes.

Perimeter of sector = arc length + 2r

Worked example: A sector has radius 8 cm and angle 45°. Find the perimeter to 3 significant figures.

Arc length = (45/360) × 2π × 8 = (1/8) × 16π = 2π ≈ 6.283 cm.

Perimeter = 6.283 + 2 × 8 = 6.283 + 16 ≈ 22.3 cm (3 s.f.).

What if the angle is given in a more complex situation?

Sometimes questions give you the arc length or area and ask you to find the radius or angle. Rearrange the formula.

Worked example — find the radius: A sector of angle 60° has arc length 5π cm. Find the radius.

Arc length = (60/360) × 2πr → 5π = (1/6) × 2πr → 5π = πr/3 → r = 15 cm.

Worked example — find the angle: A sector of radius 12 cm has area 48π cm². Find the angle.

Sector area = (θ/360) × π × 12² → 48π = (θ/360) × 144π → 48 = 144θ/360 → θ = 48 × 360/144 = 120°.

Frequently asked questions

Should I leave my answer in terms of π or use a decimal?

Read the question carefully. "Give your answer in terms of π" means leave π in the answer (e.g. 12π cm²). "Give your answer to 3 significant figures" or "to 1 decimal place" means use a decimal approximation — press π on your calculator and compute the final value. Never round π to 3.14 mid-calculation; use the full calculator value.

What is the difference between the minor arc and the major arc?

A chord divides a circle into two arcs. The minor arc is the shorter one (corresponding to the smaller angle at the centre, less than 180°). The major arc is the longer one (the angle at the centre is greater than 180°, i.e. reflex). The formulas still work for major arcs — just use the reflex angle θ.

How is the sector area formula related to the area of a full circle?

The full circle is a special sector with θ = 360°. Substituting: sector area = (360/360) × πr² = πr². The sector formula is simply the fraction of the full circle area corresponding to the fraction of the angle.

Can I use radians instead of degrees at GCSE?

No — GCSE maths uses degrees throughout. Radians appear at A-level. If you have studied radians, the formulas simplify beautifully (arc length = rθ, sector area = ½r²θ), but at GCSE always use the degree fraction method described here.

For Socratic GCSE geometry practice including circles, see aitutors.me.