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:

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:

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

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

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

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

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

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