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Area and circumference of circles: Formulae

Area and circumference of circles: Formulae

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Summary

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



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.

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 4545^\circ. The radius of the circle is 4cm4cm. What is the length of the arc and area of the shaded sector?



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}​​


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FAQs - Frequently Asked Questions

What is a sector of a circle?

What is the formula for the area of a circle?

What is the circumference of a circle?

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