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:

$\text{Area}=\pi r^2$

$\text{Circumference}=2\pi r = \pi d$

Where $r$ is the radius of the circle, and $d$ is the diameter.

Example 1

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:

$A=\pi\times r^2=\pi\times 5^2 = 25\pi$

$\underline{\text{Area}=(25\pi)m^2}$

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
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

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.

$\text{Length\,of\,an\,arc}=\frac{\theta}{360}\times\text{circumference}=\frac{\theta}{360}\times2\pi r$

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

Example 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:

$\text{Arc\,length}=\frac{\theta}{360}\times2\pi r =\frac{45}{360}\times2\times\pi\times4$

$\text{Arc\,length}=\frac{1}{8}\times8\pi=\pi$

$\underline{\text{Arc\,length}=\pi cm}$

$\text{Sector\,area}=\frac{\theta}{360}\times\pi r^2=\frac{45}{360}\times \pi\times4^2$

$\text{Sector\,area}=\frac{1}{8}\times16\pi=2\pi$

$\underline{\text{Sector\,area}=(2\pi)cm^2}$