Arc Lengths & Sector Areas

Exam code: 4024

Written by: Ashika|Reviewed by: Caroline Carroll|Updated 2 July 2026

An arc is a part of the circumference of a circle
Two points on a circumference of a circle will create two arcs
- The smaller arc is known as the minor arc
- The bigger arc is known as the major arc

A sector is the part of a circle enclosed by two radii (radiuses) and an arc
- A sector looks like a slice of a circular pizza
- The curved edge of a sector is the arc
Two radii in a circle will create two sectors
- The smaller sector is known as the minor sector
- The bigger sector is known as the major sector

You need to be able to calculate the length of an arc and the area of a sector
The angle formed in a sector by the two radii is often labelled θ (the Greek letter “theta”)
You can calculate the area of a sector or the length of an arc by adapting the formulae for the area or circumference of a circle
- A full circle is equal to 360° so the fraction will be the angle, θ°, out of 360°
  - $Area of a sector = \frac{θ}{360} \times π r^{2}$
  - $Arc length = \frac{θ}{360} \times 2 π r$

Working with sector and arc formulae is just like working with any other formula:
- Write down what you know (or what you want to know)
- Pick the correct formula
- Substitute the values in and solve

STEP 1
Divide the angle by 360 to form a fraction
- $\frac{θ}{360}$
STEP 2
Calculate the circumference of the full circle
- $2 π r$
STEP 3
Multiply the fraction by the circumference
- $\frac{θ}{360} \times 2 π r$

STEP 1
Divide the angle by 360 to form a fraction
- $\frac{θ}{360}$
STEP 2
Calculate the area of the full circle
- $π r^{2}$
STEP 3
Multiply the fraction by the area
- $\frac{θ}{360} \times π r^{2}$

Worked example

A sector of a circle is shown.

Answer:

The angle, θ, is 72° and the radius, r, is 5 cm.

(a) Find the area of the sector, giving your answer correct to 3 significant figures.

Substitute θ = 72° and r = 5 into the formula for the area of a sector, $A = \frac{θ}{360} π r^{2}$

$A = \frac{72}{360} π \times 5^{2}$

Use a calculator to work out this value

15.70796...

Round your answer to 3 significant figures

15.7 cm²

(b) Find the length of the arc of the sector, giving your answer as a multiple of $π$ .

Answer:

Substitute θ = 72° and r = 5 into the formula for the length of an arc, $l = \frac{θ}{360} 2 π r$

$l = \frac{72}{360} \times 2 \times π \times 5$

Simplify the number part without $π$

$\frac{72}{360} \times 2 \times 5 = \frac{1}{5} \times 10 = 2$

Write down the final answer with $π$

2π cm

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