12.02 Arc length and sector area

Calculate the length of a circular arc using degrees

The diagram below shows a sector of a circle. It has been formed by an angle of size $\theta$θ centered at the origin and has an arc length (the curved part of the perimeter) of length $l$l.

 

Looking at the diagram, note that a certain portion of the area of the circle has been cut out to form the sector. In particular, this is the same as the portion of the total angle that has been cut out to form $\theta$θ, and is also the same as the portion of the circumference that is the arc length $l$l.

That is, 

$\frac{\text{Area of sector}}{\text{Area of circle}}=\frac{\theta}{\text{Total angle size}}=\frac{l}{\text{Circumference of circle}}$Area of sectorArea of circle​=θTotal angle size​=lCircumference of circle​.

Now we know that the total angle measure of a circle is $360^\circ$360° and that the circumference of a circle of radius $r$r is $2\pi r$2πr, so we can rewrite the second part of this equality as

$\frac{\theta}{360^\circ}=\frac{l}{2\pi r}$θ360°​=l2πr​.

Rearranging, we can see that

$l=2\pi r\times\frac{\theta}{360^\circ}$l=2πr×θ360°​.

That is, the arc length is equal to the portion $\frac{\theta}{360^\circ}$θ360°​ of the total length of the circumference.

 

 

Practice questions

Question 1

Question 2

 

Calculate the length of a circular arc using radians

Angles in radians

We are used to measuring angles in degrees, and performing calculations with angles measured in degrees, where one degree is the size of the angle formed by dividing a full rotation into $360$360 equal portions.

Another form of angle measure is the radian. We define a radian as the size of the angle which forms an arc of equal length to the radius of the circle. This is shown in the diagram below.

 

In general, we define the size of angle in radians by the ratio of the corresponding arc length $l$l to the length of the radius $r$r. That is, we define an angle $\theta$θ in radians to be:

$\theta=\frac{l}{r}$θ=lr​

This is shown in the diagram below.

 

Calculating arc length

From the definition formula $\theta=\frac{l}{r}$θ=lr​, we can immediately obtain the rearrangement $l=r\theta$l=rθ by multiplying both sides by $r$r. This means that if we know the size of an angle in radians, we can calculate the corresponding arc length by simply multiplying the angle by the radius.

 

 

 

Practice questions

Question 3

Question 4

 

Find dimensions from the length of an arc

If we know the arc length $l$l of a sector and the radius $r$r, we can substitute these values into the appropriate formula and solve the equation for the measure of the contained angle $\theta$θ in degrees or radians.

Similarly, if we know the arc length $l$l of a sector and the value of $\theta$θ, we can substitute these values into the appropriate formula and solve for the radius $r$r.

 

Worked examples

Question 5

Solve: The sector below has a radius of $10$10 cm and an arc length of $30$30 cm. Find the measure of the contained angle $\theta$θ in degrees.

Round your answer to two decimal places.

 

Think: We want to relate the measure of the contained angle $\theta$θ in degrees with the arc length $30$30 cm and the radius $10$10 cm. We can use the formula

$l=2\pi r\times\frac{\theta}{360^\circ}$l=2πr×θ360°​

Do: Let's substitute the value of $r=10$r=10 and $l=30$l=30 into the equation. Then we can rearrange the equation to solve for $\theta$θ.

$l$l	$=$=	$2\pi r\times\frac{\theta}{360^\circ}$2πr×θ360°​	(Writing down the formula)
$30$30	$=$=	$2\pi\times10\times\frac{\theta}{360^\circ}$2π×10×θ360°​	(Substituting our values)
$\frac{30}{2\pi\times10}$302π×10​	$=$=	$\frac{\theta}{360^\circ}$θ360°​	(Dividing both sides by $2\pi\times10$2π×10)
$\frac{30\times360^\circ}{2\pi\times10}$30×360°2π×10​	$=$=	$\theta$θ	(Multiplying both sides by $360^\circ$360°)
$\theta$θ	$=$=	$171.89^\circ$171.89°	(Rounding to two decimal places)

Reflect: How might we solve for the radius $r$r, if the measure of the contained angle was $171.89^\circ$171.89° and the arc length was $30$30 cm?

 

Practice questions

question 6

question 7

Question 8

 

Find the area of a sector

The area of a sector can be found by determining what percentage of the circle the sector covers. The percentage of the circle covered is determined by its central angle, that is, the enclosed angle between the two radii.

Consider a sector with a central angle of measure $\frac{2\pi}{4}=\frac{\pi}{2}$2π4​=π2​ as shown below. A whole circle has a central angle with a measure of $2\pi$2π, so the area of the sector is one quarter of the full circle's area.

In general for a sector with an angle, measured in radians, the fraction of the area covered will be given by $\frac{\theta}{2\pi}$θ2π​.

The area of the whole circle	The area of a quarter of the circle

The area formula is derived by taking the amount of the circle covered by the sector, and multiplying by the total area of the circle.

$\text{Area}=\text{fraction of circle covered}\times\text{Area of a circle}=\frac{\theta}{2\pi}\times\pi r^2$Area=fraction of circle covered×Area of a circle=θ2π​×πr2

For the formula, in radians, we may notice that $\pi$π will cancel out. This simplified form is often used:

$A=\frac{1}{2}r^2\theta$A=12​r2θ

The formula containing $\pi$π may be easier to remember though as it shows how we derived the formula.

Worked examples

Question 9

Determine the area of the sector below in exact form.

Think: The area is given by the formula $A=\frac{1}{2}r^2\theta$A=12​r2θ and we are given $\theta$θ and $r$r.

Do: Substitute these values into the formula:

$A$A	$=$=	$\frac{1}{2}r^2\theta$12​r2θ	(Writing down the formula)
	$=$=	$\frac{1}{2}\times30^2\frac{\pi}{5}$12​×302π5​	(Substituting)
	$=$=	$\frac{900\pi}{10}$900π10​	(Simplifying the power and multiplication)
	$=$=	$90\pi$90π cm2	(Simplifying the fraction)

 

Question 10

The sector below has an area of $8$8 m² and a radius of $2$2 m. Find the value of $\theta$θ in the sector below.

Think: We are given the area and the radius of the sector so we can use the formula $\theta=\frac{2A}{r^2}$θ=2Ar2​.

Do: Substitute the values into the equation:

$\theta$θ	$=$=	$\frac{2A}{r^2}$2Ar2​	(Writing down the equation)
	$=$=	$\frac{2\times8}{2^2}$2×822​	(Substituting)
	$=$=	$\frac{16}{4}$164​	(Simplifying the top and bottom)
	$=$=	$4$4 radians	(Simplifying the fraction)

Practice questions

Question 11

Question 12