Arcs & Sectors | Edexcel GCSE Maths: Foundation Revision Notes 2017

Arc Lengths & Sector Areas

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

The angle formed in a sector by the two radii is often labelled θ (the Greek letter “theta”)
You can calculate the length of an arc by adapting the formulae for the circumference of a circle
- A full circle is equal to 360° so the portion of the circle will be the angle, θ°, out of 360°
  - $Length of an arc = \frac{θ}{360} \times 2 π r$
  - You need to remember this formulae; it is not given in the exam

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
- $\frac{θ}{360} \times 2 π r$

A sector is an area 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 can calculate the area of a sector by adapting the formulae for the area of a circle
- A full circle is equal to 360° so the portion of the circle will be the angle, θ°, out of 360°
  - $Area of a Sector = \frac{θ}{360} \times π r^{2}$
  - You need to remember this formulae; it is not given in the exam

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
- $\frac{θ}{360} \times π r^{2}$

Sector Area & Arc Length Formulae

To help remember the formulae, just remember they are a fraction of a circle's area, or a fraction of a circle's circumference

A sector of a circle is shown.

A sector

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

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

$2 π$ cm