# Areas and perimeters of sectors (degrees)

Lesson

Like we saw in our chapters describing circumferences and areas of circles we now have the following rules.

$\text{Circumference, C}=2\pi r$Circumference, C=2πr

$\text{Area, A}=\pi r^2$Area, A=πr2

What if we don't have an entire circle?

Well, half a circle would have half the area or half the circumference. One quarter of a circle would have a quarter of the area, or a quarter of the circumference.  In fact all we need to know is what fraction the sector is of a whole circle.  For this all we need to know is the angle of the sector.

Looking at the quarter circle, the angle of the sector is $90$90°. The fraction of the circle is $\frac{90}{360}=\frac{1}{4}$90360=14

More generally, If the angle of the sector is $\theta$θ, then the fraction of the circle is represented by

$fraction=\frac{\theta}{360}$fraction=θ360 (due to there being $360$360° in a circle).

#### Example

Question: Find the area and circumference of a sector with central angle of $126$126° and radius of $7$7cm. Evaluate to $2$2 decimal places.

Think: What fraction is this sector of a whole circle?  What are the rules for circumference and area?

Do:  This sector is $\frac{126}{360}=0.35$126360=0.35 of a circle.

Circumference of a whole circle is $C=2\pi r$C=2πr, so the perimeter of the sector is

 $0.35\times2\pi r$0.35×2πr $=$= $0.35\times2\pi\times7$0.35×2π×7 $=$= $0.35\times14\pi$0.35×14π $=$= $4.9\times\pi$4.9×π $=$= $15.39$15.39 cm (rounded to $2$2 decimal places)

Area of a circle is $A=\pi r^2$A=πr2, so the area of the sector is

 $0.35\times\pi r^2$0.35×πr2 $=$= $0.35\pi\times7^2$0.35π×72 $=$= $17.15\pi$17.15π $=$= $53.88$53.88 cm2  (rounded to 2 decimal places)

#### Worked Examples

##### QUESTION 1

Consider the sector below.

1. Calculate the perimeter. Give your answer correct to $2$2 decimal places.

2. Calculate the area. Give your answer correct to $2$2 decimal places.

##### QUESTION 2

Consider the sector below.

A goat is tethered to a corner of a fenced field (shown). The rope is $9$9 m long. What area of the field can the goat graze over?