Arc length and perimeter of sectors
Sectors are pieces of a circle that have been sliced out from the centre, like the ones pictured below:
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| examples of sectors | ||
The length of the the circular edge is called the arc length of the sector. The arc length of a quarter circle is a quarter of the circumference of the whole circle: $\frac{1}{4}\times2\pi r$14×2πr . A quarter circle has a contained angle of $90^\circ$90° out of a total possible $360^\circ$360° and the corresponding proportion is indeed a quarter:$\frac{90}{360}=\frac{1}{4}$90360=14. If instead we have a sector with a contained angle of $30^\circ$30°, then we have $30^\circ$30° out of a total possible $360^\circ$360°, so the fraction of the circle that we have is $\frac{30}{360}=\frac{1}{12}$30360=112 and the arc length will be $\frac{1}{12}\times2\pi r$112×2πr. This is the idea behind the arc length formula.
A sector with contained angle $\theta$θ corresponds to a fraction $\frac{\theta}{360}$θ360of a full circle and so its arc length $l$l is given by the formula:
$l=\frac{\theta}{360}\times2\pi r$l=θ360×2πr
where $r$r is the radius of the circle.
Manipulate this applet by moving the point on the circle to see the relationship between the contained angle and the arc length.
Remember that the perimeter of a 2D shape is the total distance around the boundary of the shape. For a sector the boundary consists of a curved part and two straight edges each with length equal to the radius of the circle.
When finding the perimeter of a sector, don't forget to add the lengths of the straight edges to the arc length!
| $\text{perimeter of a sector}=\text{arc length}+2r$perimeter of a sector=arc length+2r |
Practice questions
Question 1
A circle with radius $4$4 cm has been drawn with a dashed line.
A sector is outlined with a filled line.
Find the exact circumference of the whole circle.
Find the exact length of the arc of the sector.
Find the perimeter of the sector.
Round your answer to two decimal places.
Question 2
A sector has radius $6$6 cm and an angle of $55^\circ$55°.
Find the exact length of the arc.
Find the perimeter of the sector.
Round your answer to two decimal places.
Area of a sector
A sector is a fraction of a circle and so as we saw above the arc length is a fraction of the circumference. The same is true for area: the area of a sector is a fraction of the total area. For example, the area of a semicircle is half the area of the full circle.
Semicircle
$\text{Area of semicircle}=\frac{1}{2}\times\pi r^2$Area of semicircle=12×πr2
We can make a formula for the area of any sector depending on the angle $\theta$θ that subtends the arc at the centre.
A sector with contained angle $\theta$θ corresponds to a fraction $\frac{\theta}{360}$θ360of a full circle and so its area is given by:
| $A$A | $=$= | $\frac{\theta}{360}\times\pi r^2$θ360×πr2 |
Practice questions
Question 3
Consider the shape below:
What fraction of the circle is shown?
Find the exact area of a circle with a radius of $3$3 cm.
Now, find the exact area of the given shape.
How could we find the area of this part of the circle in just one step?
That is, what formula could we use to find the area?
$\pi r^2$πr2
A$\pi r^2+\frac{1}{2}$πr2+12
B$2\pi r^2$2πr2
C$\frac{1}{2}\pi r^2$12πr2
D
Question 4
Find the area of the following sector of a circle.
Round your answer to one decimal place.
