12. Circles

Lesson

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=`θ`Total angle size=`l`Circumference 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$2π`r`, so we can rewrite the second part of this equality as

$\frac{\theta}{360^\circ}=\frac{l}{2\pi r}$`θ`360°=`l`2π`r`.

Rearranging, we can see that

$l=2\pi r\times\frac{\theta}{360^\circ}$`l`=2π`r`×`θ`360°.

That is, the arc length is equal to the portion $\frac{\theta}{360^\circ}$`θ`360° 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`=2π`r`×`θ`360°.

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

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.

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`=`l``r`

This is shown in the diagram below.

From the definition formula $\theta=\frac{l}{r}$`θ`=`l``r`, we can immediately obtain the rearrangement $l=r\theta$`l`=`r``θ` 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`=`l``r`.

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

Careful!

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

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.

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.

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`.

**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`=2π`r`×`θ`360°

**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?

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.

Round your answer to two decimal places.

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.

Round your answer to two decimal places.

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.

Round your answer to two decimal places.

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 $\frac{\theta}{2\pi}$`θ`2π.

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=`θ`2π×π`r`2

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`r`2`θ`

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`=`θ`2π×π`r`2=12`r`2`θ`

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`r`2`θ` and we are given $\theta$`θ` and $r$`r`.

**Do:** Substitute these values into the formula:

$A$A |
$=$= | $\frac{1}{2}r^2\theta$12r2θ |
(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 cm^{2} |
(Simplifying the fraction) |

The sector below has an area of $8$8 m^{2} 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`=2`A``r`2.

**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) |

The sector below has an area of $3.4$3.4 cm^{2} and a radius of $2$2 cm.

Find the value of $\theta$

`θ`in radians.

The area $A$`A` of the minor sector $OAC$`O``A``C` shown below is $252\pi$252π cm^{2}. The original circle had a center at $O$`O` and a radius $r$`r` measuring $42$42 cm.

Find the value of $\theta$

`θ`, the measure of the angle $\angle AOC$∠`A``O``C`.Give your answer in radians in terms of $\pi$π.

Find the exact length of the arc $AC$

`A``C`.Give your answer in terms of $\pi$π.

Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.