Question

If the perimeter of a certain sector is 10 units and radius of the circle is 3 units , find the area of sector ?

IIT { practice set }

Answer 1

Given,

Perimeter of a sector = 10 units

Length of arc = 10 units - 6 units = 4 units.

Radius of circle = 3 units.

Area of sector = ?

Fitst find the circumference of circle.

Circumference of circle = 2 π r

= 2 × ( 22/7 ) × 3

= ( 132 / 7 ) Units.

Now,

Central angle / 360°=length of arc /Circumference of circle

●/ 360° = 4 / ( 132 / 7 )

●/ 360° = 28 / 132

● = ( 28 × 360° ) / 132

● = ( 14 × 360° ) / 66

● = ( 14 × 180° ) / 33

● = ( 14 × 60° ) / 11

● = 840° / 11.

Area of circle = πr^2

= ( 22 / 7 ) × ( 3 units )^2

= ( 22 / 7 ) × 9 units^2

= ( 198 / 7 ) units^2

Area of sector = ( ● / 360° ) × ar. of circle

= { ( 840° / 11 ) / 360° } × ( 198 / 7 ) unit^2

= { 840° / ( 360° × 11 ) } × ( 198 / 7 ) unit^2

= { 7 / 33 } × ( 198 / 7 ) unit^2

= ( 198 / 33 ) unit^2

= 6 unit^2.

Answer 2

Radius of circle = 3 units

Perimeter of an arc = 10 units

Length of that arc = Perimeter of that arc - 2 × Radius  of that circle

                             = 10 units - 2 × 3 units

                            = 10 units - 6 units = 4 units.

Perimeter of circle = 2πr

                              = 2 ( 22/7 ) 3 unit

                              = ( 132 / 7 ) unit
Area of circle = πr²

                     = ( 22/7 ) × ( 3 units )²

                    = ( 22/7 ) × 9 unit²

                    = ( 198 / 7 ) unit²

To find the area of a sector we have a formula,

Area of a sector = ( Central angle / 360° ) × Area of circle

Now,

Let the central angle is α.


⇒ ( 360° / α ) = Perimeter of circle ÷ Length of Arc

⇒ ( 360° / α ) = ( 132 / 7 ) unit ÷ 4 unit

⇒ ( 360° / α ) = ( 132 / 7 × 4 )

⇒ ( 360° / α ) = ( 33 / 7 )

⇒ α = ( 360° × 7 ) ÷ 33

⇒ α = ( 120° × 7 ) ÷ 11

∴ α = 840° / 11.

Now,

Area of sector = ( α /  360° ) × Area of circle

                        = { ( 840° / 11 ) ÷ 360° } × ( 198 / 7 ) unit²

                        = { 840° / 11 × 360° } × ( 198 / 7 ) unit²

                        =  (7 / 11 × 3 ) × ( 198 / 7 ) unit²

                        = ( 7 / 33 ) × ( 198 / 7 ) unit²

                        = ( 198 / 33 ) unit²

                        = 6 unit²

The required answer is 6 unit².

Hope it helps !!