Arc Length, Area of a Circular Sector
Arc of a circle:
An arc is a part of the circumference of a circle. The red part AB in the figure below shows an arc.
Note: The length ''l'' of an arc that subtends an angle `\theta` at the center of a circle of radius r is given by
l = r`\theta`
i.e. Arc length is angle times radius of the circle. (Here `\theta` is in radians, not in degrees)
Example 1:
The radius of a circle is r = 10 cm. Find the length of an arc that subtends an angle of 1.4 radians at the center of the circle.
Solution: We are given that r = 10 cm
`\theta` = 1.4 radians
=> Arc Length = l = r.`\theta`
= (10cm)(1.4)
= 14cm
Example 2:
A cyclist is moving along a circular path of radius 18m. How much distance does he travel if he makes a total of 4 revolutions?
Solution: We know that 1 revolution = 2`\pi` radians
=> 4 revolutions = 4 x 2`\pi` radians = 8`\pi` radians
Since the cyclist moves along the circular path, he moves along a trajectory that is part of the circumference of the circle (i.e. an arc) as shown in the figure.
=> Distance travelled = arc length = radius x angle in radians
=> Distance travelled = (18m) (8`\pi`) = 1448`\pi` m
Segment of a Circle:
A segment is the part of a circle bounded by an arc and a chord of the circle. The shaded part in the following figure indicates the segment of the circle.
Sector of a Circle:
A sector is the part of a circle bounded by two radii and an arc of the circle as shown in the figure. The part shown in maroon is a sector of the circle.
Area of a circular sector
Let r be the radius of a circle and let there be an arc of length l on that circle that subtends an angle `\theta` at the center of the circle.
We know that,
The area of the circle = `\pir^2`
Angle of the circle = 2`\pi` radians
Angle of sector POQ = `\theta`
Then according to the elementary geometry
`(text(Area of Sector) POQ)/(text(Area of Circle))` = `(text(Angle of sector))/(text(Angle of Circle))`
`(text(Area of Sector) POQ)/(\pir^2 )` = `\theta/( 2\pi )`
=> Area of Sector POQ = `\theta/( 2\pi ) ` x `\pi r^2`
=> Area of Sector POQ = `1/( 2 ) \theta r^2`
Example 3:
Find area of the sector having a central angle of 20° in a circle of radius 7m.
Solution: We are given that r = 7m
`\theta = 20^\circ` x `\pi/180` = `\pi/( 9 )` radians
Area of sector = `( 1 )/2 r^2 \theta`
= `( 1 )/2 (7)^2 (\pi/( 9 ))`
= `( 1 )/2 (49) (\pi/( 9 ))`
since, (`\pi` = 3.1416)
= `( 49\pi )/( 18) m^2`
Example 4:
Imagine a circular track of radius 500m and let a train be running on this track at a rate of 30km per hour. Calculate the distance covered by the train in a time interval of 10 seconds. Also find the angle through which it turns in this interval.
Solution: We are given that r = 500m
v = 30km/hr
=> Distance travelled in 1 second = 30 x 1000 x `1/( 3600 )` m
= `( 25 )/3` m = 8.33m
=> Distance travelled in 10 seconds = l = `( 250 )/3` m = 83.3m
Therefore, The train travels 83.3m in 10 seconds.
Let `\theta` be the angle through which the train turns in 10 seconds.
=> `\theta` = `( l )/r` = `( 250 )/3` ÷ 500
=> `\theta` = `( 1 )/6` radian
Therefore, The train turns through an angle of `1/6` radian in the interval of 10 seconds.