How to Find the Length of Arc and the Area of Sector
Read,2 minutes
What is the Area of the Sector of a Circle?
The area of a sector in the region that a circle's sector encloses. Minor and major sectors are the two different categories of sectors. A major sector is larger than a semi-circle, whereas a minor sector is smaller than a semi-circle.
![Sector2](/Images/Article/Sector(61646).png)
Sector Area Formula
We use the area of a sector formula to get the overall area that the sector encompasses. The following formulae can be used to determine a sector's area:
Area of a Sector of a Circle \(= \ (\frac{θ}{360}) \times πr^2\), where \(r\) is the circle's radius and \(θ\) is the sector angle, in degrees, that the arc at the center subtends.
Area of a Sector of a Circle \(= \ \frac{1}{2} \times r^2θ\), where \(r\) is the circle's radius and \(θ\) sector angle, expressed in radians that the arc at the circle's center subtends.
Arc Length
Any section of a circle's circumference is considered an arc. The length of an arc is the distance between its two endpoints. Knowing a little bit about circle geometry is necessary to determine an arc length. Since the arc is a component of the circumference, you can determine the arc length if you know what percentage of \(360\) degrees the arc's central angle is.
![Arc](/Images/Article/Arc(23216).png)
Arc Length Formula
Arc length \(= \ \frac{Central \ Angle}{360°} \times 2πr\)
Example
Find the length of the red arc. (\(π \ = \ 3.14\))
![Arc Length](/Images/Article/Arc_Length(2096).png)
Solution
Arc length \(= \ \frac{Central \ Angle}{360°} \times 2πr\)
As shown in the above picture, \(r \ = \ 9\) and Central Angle \(= \ 150°\), So:
Arc length \(= \ \frac{150°}{360°} \times 2π \times 9\) \(= \ \frac{5}{12} \times 2(3.14) \times 9 \ = \ 23.55\)
Therefore, the arc length is \(23.55\).
Free printable Worksheets
Exercises for Finding the Arc Length and the Sector Area
1) Find the arc length.
![Arc Length](/Images/Article/Arc_Length1(23710).png)
2) Find the arc length.
![Arc Length2](/Images/Article/Arc_Length2(40145).png)
3) Find the arc length.
![Arc Length3](/Images/Article/Arc_Length3(87555).png)
4) Find the arc length.
![Arc Length](/Images/Article/Arc_Length4(9034).png)
5) Find the arc length.
![Arc Length](/Images/Article/Arc_Length5(7381).png)
6) Find the sector area.
![Arc Length6](/Images/Article/Arc_Length6(22250).png)
7) Find the sector area.
![Arc Length](/Images/Article/Arc_Length7(90433).png)
8) Find the sector area.
![Arc Length8](/Images/Article/Arc_Length8(3618).png)
9) Find the sector area.
![Arc Length9](/Images/Article/Arc_Length9(77342).png)
10) Find the sector area.
![Arc Length10](/Images/Article/Arc_Length10(73929).png)
1) Find the arc length.
![Arc Length](/Images/Article/Arc_Length1(23710).png)
\(\color{red}{arc \ length \ = \ \frac{150}{360} \times 2π(3) \ = \ \frac{5π}{2} \ ≈ \ 7.85}\)
2) Find the arc length.
![Arc Length2](/Images/Article/Arc_Length2(40145).png)
\(\color{red}{arc \ length \ = \ \frac{136}{360} \times 2π(5) \ = \ \frac{34π}{9} \ ≈ \ 11.862}\)
3) Find the arc length.
![Arc Length3](/Images/Article/Arc_Length3(87555).png)
\(\color{red}{arc \ length \ = \ \frac{160}{360} \times 2π(5) \ = \ \frac{40π}{9} \ ≈ \ 13.955}\)
4) Find the arc length.
![Arc Length](/Images/Article/Arc_Length4(9034).png)
\(\color{red}{arc \ length \ = \ \frac{250}{360} \times 2π(9) \ = \ \frac{25π}{2} \ ≈ \ 39.25}\)
5) Find the arc length.
![Arc Length](/Images/Article/Arc_Length5(7381).png)
\(\color{red}{arc \ length \ = \ \frac{330}{360} \times 2π(12) \ = \ 22π \ ≈ \ 69.08}\)
6) Find the sector area.
![Arc Length6](/Images/Article/Arc_Length6(22250).png)
\(\color{red}{sector \ area \ = \ \frac{80}{360} \times π(6)^2 \ = \ 8π \ ≈ \ 25.12}\)
7) Find the sector area.
![Arc Length](/Images/Article/Arc_Length7(90433).png)
\(\color{red}{sector \ area \ = \ \frac{115}{360} \times π(4)^2 \ ≈ \ 16.048}\)
8) Find the sector area.
![Arc Length8](/Images/Article/Arc_Length8(3618).png)
\(\color{red}{sector \ area \ = \ \frac{340}{360} \times π(3)^2 \ = \ \frac{17π}{2} \ ≈ \ 26.69}\)
9) Find the sector area.
![Arc Length9](/Images/Article/Arc_Length9(77342).png)
\(\color{red}{sector \ area \ = \ \frac{60}{360} \times π(4)^2 \ = \ \frac{8π}{3} \ ≈ \ 8.373}\)
10) Find the sector area.
![Arc Length10](/Images/Article/Arc_Length10(73929).png)
\(\color{red}{sector \ area \ = \ \frac{170}{360} \times π(6)^2 \ = \ 17π \ ≈ \ 53.38}\)
Arc Length and Sector Area Practice Quiz