New York Geometry - 2020 Edition
10.02 Arc length and sector area
Lesson

## 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.

A sector of a circle with arc length $l$l, formed by an angle of $\theta$θ.

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.

Summary

In a circle of radius $r$r, we can find the length of an arc $l$l formed by an angle of $\theta$θ by using the formula $l=2\pi r\times\frac{\theta}{360^\circ}$l=2πr×θ360°.

#### Practice questions

##### Question 1

If the circumference of the circle is equal to $144$144 cm, find the length of the solid arc.

##### Question 2

A sector of a circle of radius $7$7 cm is formed from an angle of size $41^\circ$41°.

Find the length of the arc rounded to two decimal places.

## Calculate the length of a circular arc using 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.

An angle of one radian forms an arc of equal length to the radius of the circle.

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.

An angle $\theta$θ in radians is equal to the ratio $\frac{l}{r}$lr.

### 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.

Summary

The size of an angle $\theta$θ in radians is defined by the ratio of the arc length $l$l to the radius $r$r. That is, $\theta=\frac{l}{r}$θ=lr.

We can rearrange this into the form $l=r\theta$l=rθ in order to calculate the arc length for a given angle.

Careful!

The formula $l=r\theta$l=rθ can only be used to calculate arc length for an angle $\theta$θ that is measured in radians.

#### Practice questions

##### Question 3

The diagram shows a sector of a circle of radius $5$5 units, formed from an angle of size $2.3$2.3 radians.

Find the exact length of the arc.

##### Question 4

A sector of a circle of radius $7$7 m is formed from an angle of size $\frac{5\pi}{4}$5π4 radians.

Find the exact length of the arc.

## 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.

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

The sector below has a contained angle with a measure of $115^\circ$115° and an arc length of $23$23 cm. Find the radius $r$r of the sector in centimeters.

##### question 7

A sector has a radius of $2$2 cm and an arc length of $5$5 cm. Find the measure of the contained angle $\theta$θ in degrees.

##### Question 8

The sector below has a contained angle with a measure of $\frac{2\pi}{3}$2π3 in radians and an arc length of $23$23 cm. Find the radius $r$r of the sector in centimeters.

## 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=12r2θ

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

Area of a sector

A sector with a radius $r$r and an angle with a measure $\theta$θ, in radians, will have an area given by the formula:

$\text{Area}=\text{fraction of circle covered}\times\text{Area of a circle}$Area=fraction of circle covered×Area of a circle

That is:

$A=\frac{\theta}{2\pi}\times\pi r^2=\frac{1}{2}r^2\theta$A=θ2π×πr2=12r2θ

#### 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=12r2θ 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\times4}{2^2}$2×422​ (Substituting) $=$= $\frac{8}{4}$84​ (Simplifying the top and bottom) $=$= $2$2 radians (Simplifying the fraction)

#### Practice questions

##### Question 11

The sector below has an area of $3.4$3.4 cm2 and a radius of $2$2 cm.

1. Find the value of $\theta$θ in radians.

##### Question 12

The area $A$A of the minor sector $OAC$OAC shown below is $252\pi$252π cm2. The original circle had a center at $O$O and a radius $r$r measuring $42$42 cm.

1. Find the value of $\theta$θ, the measure of the angle $\angle AOC$AOC.

Give your answer in radians in terms of $\pi$π.

2. Find the exact length of the arc $AC$AC.

Give your answer in terms of $\pi$π.

### Outcomes

#### GEO-G.CO.1

Know precise definitions of angle, circle, perpendicular lines, parallel lines, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc as these exist within a plane.

#### GEO-G.C.5

Using proportionality, find one of the following given two others; the central angle, arc length, radius or area of sector.