Register Login
Evulpo Logo

Benefit from a yearly subscription now!

Save 44%

£11.99 / Month

£6.67 / Month

Chapter overview

Maths

Exam board

AQA

Number

Algebra

Graphs

Ratio proportion and rates of change

Shapes and area

Angles and geometry

Trigonometry

Probability

Statistics

Maths

Area and circumference of circles: Formulae

Your lesson progress
 
 
0%

Summary

Download

Area and circumference of circles: Formulae

In a nutshell

There are separate formulae for the area and perimeter of a circle. These formulae can also be applied to find the length of arcs and area of sectors.



Area and circumference of a circle

The circumference is the name given to the perimeter of the circle.

The area and circumference of a circle are given by the formulae:


​Area=πr2\text{Area}=\pi r^2​

Circumference=2πr=πd\text{Circumference}=2\pi r = \pi d​​


Where rr​ is the radius of the circle, and dd​ is the diameter.


Example 1

What is the area and perimeter of a circle that has a diameter of 10m10m?


Substitute d=10d=10 into the formula for the circumference:

C=πd=10πC=\pi d=10\pi​​


​Circumference=(10π)m‾\underline{\text{Circumference}=(10\pi)m}​​


For the area, first find the radius by halving the diameter:

​r=d÷2=10÷2=5r=d\div2=10\div2=5​​


Substitute r=5r=5 into the formula for the area:

A=π×r2=π×52=25πA=\pi\times r^2=\pi\times 5^2 = 25\pi​​


​Area=(25π)m2‾\underline{\text{Area}=(25\pi)m^2}​​



Labelling parts of a circle

These are the main parts of a circle you need to know:


name

description

illustration

Arc

A curved line that is part of the circumference


Maths; Shapes and area; KS4 Year 10; Area and circumference of circles: Formulae

Sector

A portion of the circle that is divided by two radii (plural of radius)
Maths; Shapes and area; KS4 Year 10; Area and circumference of circles: Formulae

Chord

A line that joins two points on the circumference
Maths; Shapes and area; KS4 Year 10; Area and circumference of circles: Formulae

Segment

A part of the circle that is enclosed by a chord and an arc
​
Maths; Shapes and area; KS4 Year 10; Area and circumference of circles: Formulae



Area and perimeter of arcs and sectors

It's possible to apply the formula for the area and circumference of a circle to arcs and sectors.

Length of an arc=θ360×circumference=θ360×2πr\text{Length\,of\,an\,arc}=\frac{\theta}{360}\times\text{circumference}=\frac{\theta}{360}\times2\pi r​​


​Area of a sector=θ360×area of circle=θ360×πr2\text{Area\,of\,a\,sector}=\frac{\theta}{360}\times\text{area\,of\,circle}=\frac{\theta}{360}\times\pi r^2​​


Example 2

In the diagram below, the angle between the two radii is 45∘45^\circ. The radius of the circle is 4cm4cm. What is the length of the arc and area of the shaded sector?


Maths; Shapes and area; KS4 Year 10; Area and circumference of circles: Formulae


Substitute r=4r=4 and θ=45\theta=45 into both equations:

​Arc length=θ360×2πr=45360×2×π×4\text{Arc\,length}=\frac{\theta}{360}\times2\pi r =\frac{45}{360}\times2\times\pi\times4​​


​Arc length=18×8π=π\text{Arc\,length}=\frac{1}{8}\times8\pi=\pi​​


​Arc length=πcm‾\underline{\text{Arc\,length}=\pi cm}​​


​Sector area=θ360×πr2=45360×π×42\text{Sector\,area}=\frac{\theta}{360}\times\pi r^2=\frac{45}{360}\times \pi\times4^2​​


​Sector area=18×16π=2π\text{Sector\,area}=\frac{1}{8}\times16\pi=2\pi​​


​Sector area=(2π)cm2‾\underline{\text{Sector\,area}=(2\pi)cm^2}​​