Areas Related to Circles

Length of Arc and Area of Sector of Circle

Some important formulae to remember are:

In case of quadrant, θ = 90°. Therefore,

In case of semicircle, θ = 180°. Therefore,

Relation between the area of sector and the length of an arc:
Look at the following figure.

Here, O is the centre of the circle of radius r and APB is an arc of length l.

Clearly, measure of arc APB = Central angle = θ

Let the area of sector OAPB be A.

Now, we have


 From length of arc and area of corresponding sector, we can also conclude that.

These formulae are very helpful to calculate various attributes related to circle. Let us solve some examples to learn the concept better.
Example 1: Find the area of the sectors OACB and OADB. Also find the length of the arc ACB. (Take π =    )

Solution:

Given, radius of the circle, r = 6.3 cm 

Angle of sector OACB, θ = 40°

θ360×πr2


 = 13.86 cm2

And, area of sector OADB = πr2 − area of the sector OACB


= 124.74 − 13.86 cm2

= 110.88 cm2

Length of arc ACB =  

= 4.4 cm

Example 2: Find the area swept by the hour hand of length 10 cm of a clock when it moves from 2 p.m. to 4 p.m. Also find the length covered by the tip of the hour hand during that time. (Take π =3.14)

Solution: 


When an hour hand moves from 2 p.m. to 4 p.m., it moves for 2 hours.

The hour hand takes 12 hours to complete 1 rotation i.e., to rotate 360°.

In 1 hour, it will rotate = 

In 2 hours, it will rotate = 30° × 2 = 60°

Therefore, when it moves from 2 p.m. to 4 p.m., it sweeps an area equal to the area of a sector of angle 60°.
 
Radius of this sector = length of the hour hand = 10 cm


∴ Area swept 

= 52.33 cm2 (approx.)


Length covered by the tip 

= 10.47 cm (approx.)

Example 3: If the area and length of the arc of a sector are   cm2 and   cm respectively, then find out the radius of the circle and the angle of sector.

Solution:

Let r be the radius of the circle and θ be the angle of sector.


It is given that the area of the sector is  cm2.

Also, the length of the arc is  cm.

⇒   = … (2)
On dividing Equation (1) with Equation (2), we obtain


cm
 
On putting the value of r in equation (1), we obtain

   =  

Thus, the radius of the circle is 16 cm and the angle of the sector is 105°.

Example 4: Find the area of the sector OAXB and length of the arc AXB for the given figure.

Solution:

In the figure, OA = AB = OB (All are of length 7 cm) Hence, ΔOAB is an equilateral triangle.
∴ 

⇒ θ = 60°


Area of the sector =  cm2


Length of the arc =  cm

Example 5: A truck has two wipers which do not overlap and each wiper has a blade of length 28 cm sweeping through an angle of 90°. What is the total area cleaned at each sweep of the blades?
 

Solution:

It is clear that each wiper sweeps a quadrant of a circle of radius 28 cm.

Area cleaned at each sweep =  

Example 6: Radius of a circle is 14 cm. What is the area of the sector corresponding to the arc of length 20 cm?

Solution:

Radius of circle (r) = 14 cm

Length of arc (l) = 20 cm

Example 7: Area of a sector of a circle whose radius is 12 cm is 132 cm2. Find the measure of central angle and length of the corresponding arc.

Solution:

Area of sector (A) = 132 cm2

Radius of circle (r) = 12 cm
Let the measure of central angle be θ and length of arc be l.

Now, we have

Thus, the central angle measures 105°.

Also, we have