Interesting isn't it that when we realign the segments we end up with a parallelogram shape. Which is great, because it means we know how to find the area based on our knowledge that the area of a parallelogram has formula $A=bh$A=bh. In a circle, the base is half the circumference and the height is the radius.

Area of a Circle

$\text{Area of a circle}=\pi r^2$Area of a circle=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 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 is the rule for area?

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

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

More Worked Examples

Question 1

If the radius of the circle is $5$5 cm, find its area.

Give your answer as an exact value.

Question 2

Find the area of the shaded region in the following figure, correct to one decimal place.

Question 3

Find the area of the shaded region in the following figure, correct to one decimal place.

A figure of a square with a quarter-circle cut out of it such that the bottom-right corner of the square intersects with the corner of the quarter-circle. The bottom-right corner has a small square indicating a right angle. The top and left sides of the square each has a tick mark. The right side of the square is divided into two segments with $4$4-cm segment on top of the $20$20-cm segment. The $20$20-cm segment is also the radius of the quarter-circle.

Question 4

Consider the sector below.

Calculate the perimeter. Give your answer correct to two decimal places.
Calculate the area. Give your answer correct to two decimal places.

Question 5

Consider the sector below.

A sector of a circle with a central angle marked with $59^\circ$59°, as indicated by the shaded arc. The radius is labeled as $72.2$ $cm$ .

Calculate the perimeter. Round your answer to two decimal places.
Calculate the area. Round your answer to two decimal places.

QUESTION 6

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?

Give your answer correct to two decimal places.

Area of Circles and Sectors - mixed questions