## About "How to find the area of sector"

How to find the area of sector ?

We have two methods to find the area of a sector.

(i) If central angle and radius is given

(ii) If length of arc and radius is given

Area of sectors is the region bounded by the bounding radii and the arc of the sector.

Area of a sector = (θ/360) x Π r ² square units

(or)  Area of the sector = (l r/2) square units

θ - central angle formed by the sector

L - length of arc

r - radius of the sector

We can use the first formula if the central angle(θ) formed by the sector and radius given. If the length of arc(L) is given we have to use the second formula.

## Area of sector when angle is given

Question 1 :

Find the area of the sector whose radius and central angle are 42 cm and 60° respectively.

Solution :

Area of the sector = (θ/360) x Π r²

r = 42 cm , θ = 60°

By applying those values in the above formula we get,

=  (60/360) x (22/7) x 42 x 42

= (1/6) x 22 x 6 x 42 ==> 924 cm²

Hence, the required area is 924 cm²

Question 2 :

Find the area of the sector whose radius and central angle are 21 cm and 60° respectively.

Solution:

Area of the sector = (θ/360) x Π r²

r = 21 cm , θ = 60°

By applying those values in the above formula we get,

=  (60/360) x (22/7) x 21 x 21

= (1/6) x 22 x 3 x 21 ==> 231 cm²

Hence, the required ares is 231 cm²

Question 3 :

Find the area of the sector whose radius and central angle are 4.9 cm and 30° respectively.

Solution :

Area of the sector = (θ/360) x Π r²

r = 4.9 cm , θ = 30°

By applying those values in the above formula we get,

=  (30/360) x (22/7) x 4.9 x 4.9

= (1/12) x 22 x 0.7 x 4.9 ==> 6.3 cm²

Hence, the required area is 6.3 cm²

## Area of sector when length of arc is given

Question 4 :

Find the area of the sector and also find the central angle formed by the sector whose radius is 21 cm and length of arc is 66 cm.

Solution :

Area of the sector = (l r / 2) square units

L = 66 cm

r = 21 cm

= (1/2) x 66 x 21 ==> 33 x 21  ==> 693 square units

Area of the sector = 693 square units

(θ/360) x Π r ² = 693

(θ/360) x (22/7) x (21)² = 693

θ = (693 x 7 x 360) / (22 x 21 x 21)

θ = (693 x 360) / (22 x 3 x 21)

θ = (231 x 180) / (11 x 21)

θ = (21 x 180) / 21

θ = 180°

Hence, area of sector and central angle are 693 square units and 180° respectively.

Question 5 :

Find the area of the sector whose radius and length of arc are 6 cm and 20 cm.

Solution :

Area of the sector = (Lr/2) square units

L = 20 cm

r = 6 cm

= (20 x 6)/2 ==> 10 x 6  ==> 60 cm²

Hence, area of sector is 60 cm²

Question 6 :

Find the area of the sector whose diameter and length of arc are 10 cm and 40 cm.

Solution :

Area of the sector = (Lr/2) square units

L = 40 cm

diameter = 10 ==> r = 5 cm

= (40 x 5)/2 ==> 20 x 5  ==> 100 cm²

Hence, area of sector is 100 cm²

## How to find the area from the given perimeter ?

Question 7 :

Find the area of the sector whose radius is 35 cm and perimeter is 147 cm.

Solution :

Perimeter of sector = 147 cm

L + 2r  =  147

L + 2 (35) = 147

L + 70 = 147

L = 77 cm

Now we have the length of an arc and radius. So we have to use the second formula to find the area of the given sector.

Area of the sector = (1/2) x l r square units

= (1/2) x 77 x 35 ==> 38.5 x 35 ==> 1347.5 square units

Question 8 :

Find the area of the sector whose radius is 20 cm and perimeter is 110 cm.

Solution :

Perimeter of sector = 110 cm

L + 2r  =  110

L + 2 (20) = 110

L + 40 = 110

L = 70 cm

Now we have the length of an arc and radius. So we have to use the second formula to find the area of the given sector.

Area of the sector = (l r/2) square units

=  (70 x 20) / 2 ==> 70 x 10 ==> 700 square units

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