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Chapter overview
Learning goals
Learning Goals
Maths
Summary
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.
The circumference is the name given to the perimeter of the circle.
The area and circumference of a circle are given by the formulae:
$\text{Area}=\pi r^2$
$\text{Circumference}=2\pi r = \pi d$
Where $r$ is the radius of the circle, and $d$ is the diameter.
What is the area and perimeter of a circle that has a diameter of $10m$?
Substitute $d=10$ into the formula for the circumference:
$C=\pi d=10\pi$
$\underline{\text{Circumference}=(10\pi)m}$
For the area, first find the radius by halving the diameter:
$r=d\div2=10\div2=5$
Substitute $r=5$ into the formula for the area:
$A=\pi\times r^2=\pi\times 5^2 = 25\pi$
$\underline{\text{Area}=(25\pi)m^2}$
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 

Sector  A portion of the circle that is divided by two radii (plural of radius)  
Chord  A line that joins two points on the circumference  
Segment  A part of the circle that is enclosed by a chord and an arc  
It's possible to apply the formula for the area and circumference of a circle to arcs and sectors.
$\text{Length\,of\,an\,arc}=\frac{\theta}{360}\times\text{circumference}=\frac{\theta}{360}\times2\pi r$
$\text{Area\,of\,a\,sector}=\frac{\theta}{360}\times\text{area\,of\,circle}=\frac{\theta}{360}\times\pi r^2$
In the diagram below, the angle between the two radii is $45^\circ$. The radius of the circle is $4cm$. What is the length of the arc and area of the shaded sector?
Substitute $r=4$ and $\theta=45$ into both equations:
$\text{Arc\,length}=\frac{\theta}{360}\times2\pi r =\frac{45}{360}\times2\times\pi\times4$
$\text{Arc\,length}=\frac{1}{8}\times8\pi=\pi$
$\underline{\text{Arc\,length}=\pi cm}$
$\text{Sector\,area}=\frac{\theta}{360}\times\pi r^2=\frac{45}{360}\times \pi\times4^2$
$\text{Sector\,area}=\frac{1}{8}\times16\pi=2\pi$
$\underline{\text{Sector\,area}=(2\pi)cm^2}$
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.
The circumference is the name given to the perimeter of the circle.
The area and circumference of a circle are given by the formulae:
$\text{Area}=\pi r^2$
$\text{Circumference}=2\pi r = \pi d$
Where $r$ is the radius of the circle, and $d$