4.06 Area of circles and sectors

Area of a circle

To find the area of a circle, we need to know its radius, $r$r. The area, $A$A,  can be found using the formula:

Area	$=$=	$\pi\times\text{radius }^2$π×radius 2
$A$A	$=$=	$\pi r^2$πr2

 

Area of a sector

Let's start with the area of a circle being equal to $\pi r^2$πr2. Remember when we found the perimeter of a sector, we found the fraction of the total circumference for the arc length. It makes sense, then, that the area of a sector is some fraction of that full area. Sometimes it is easy to recognise, for instance a semicircle is half of the total area. Or maybe you might recognise a quarter, a third, or three quarters quite easily.

Semicircle

$\text{Area of semicircle}=\frac{1}{2}\times\pi r^2$Area of semicircle=12​×πr2

 

We can make a formula for any fraction depending on the angle, $\theta$θ, that subtends the arc at the centre.

 

Minor Sector	Major Sector
$A=\frac{\theta}{360^\circ}\times\pi r^2$A=θ360°​×πr2	$A=\frac{\theta}{360^\circ}\times\pi r^2$A=θ360°​×πr2

 

The fraction at the beginning of the formula, $\frac{\theta}{360^\circ}$θ360°​, is the proportion of $360^\circ$360° that the sector's angle at the centre takes up.

Remember that a minor sector is smaller than a semi-circle, that is, when $\theta<180^\circ$θ<180°. A major sector is bigger than a semi-circle, that is, when $\theta>180^\circ$θ>180°.

 

 

Practice questions

Question 1

Question 2

Question 3

 

Areas of composite shapes involving circles 

If we combine our understanding of the area of circles and sectors with the area other shapes, we can add or subtract pieces to find unusually shaped areas. We can also map restricted areas that form a circle or sector using a fixed centre and a radius that is determined by some maximum distance. The questions below step through some strategies to solve these sorts of problems.

 

Practice questions

Question 4

Question 5

Question 6