Question

The perimeter of a sector of the circle
of area 251 sq.cm is 20 cm. Find the
area of the sector.
​

Answer 1

» To Find :

The Area of the Sector.

» Given :

• Area of the Circle = 251 cm²

• Perimeter of the Sector = 29 cm

» We Know :

Area of a Circle :

$\sf{\underline{\boxed{A_{Circle} = \pi r^{2}}}}$

Perimeter of a Sector :

$\sf{\underline{\boxed{P_{sector} = \it{l} + 2r}}}$

Area of a Sector :

$\sf{\underline{\boxed{A_{Sector} = \dfrac{\it{l}r}{2}}}}$

Where ,

• r = Radius

• l = Slant height

• P = Perimeter

• A = Area

» Concept :

According to the question ,the sector was taken from the circle of area 251 cm² ,so the radius of the Circle will be Equal to the radius of the Sector . i.e,

$\therefore$ Radius of the Circle = Radius of the sector .

For finding the Radius :

Given :

• Area = 251 cm²

Formula :

$\sf{\underline{\boxed{A_{Circle} = \pi r^{2}}}}$

Substituting the values in it ,we get :

$\sf{\Rightarrow 251 = \pi r^{2}}$

Taking , π = 22/7

$\sf{\Rightarrow 251 = \dfrac{22}{7} r^{2}}$

$\sf{\Rightarrow 251 \times 7 = 22 r^{2}}$

$\sf{\Rightarrow 1757 = 22 r^{2}}$

$\sf{\Rightarrow \dfrac{1757}{22} = r^{2}}$

$\sf{\Rightarrow 79.86 = r^{2}}$

$\sf{\Rightarrow \sqrt{79.86} = r}$

$\sf{\Rightarrow 8.94 = r}$

Hence , the Radius of the Circle is 8.94 cm.

We also know that ,the radius of the Circle is equal to the Radius of the Sector.

So ,we get :

$\sf{\underline{\boxed{R_{Circle} = R_{sector} = 8.94}}}$

Hence ,the radius of the Sector is 8.94.

Now by this information ,we can find the Slant height of the Sector ,by using the formula for Perimeter of a Sector.

» Solution :

Slant height of a Sector :

Given :

• Perimeter = 20 cm

• Radius = 8.94 cm

Formula :

$\sf{\underline{\boxed{P_{sector} = \it{l} + 2r}}}$

Substituting the values in it, we get :

$\sf{\Rightarrow 20 = \it{l} + 2 \times 8.94}$

$\sf{\Rightarrow 20 = \it{l} + 17.88}$

$\sf{\Rightarrow 20 - 17.88 = \it{l}}$

$\sf{\Rightarrow 2.12 cm = \it{l}}$

Hence ,the Slant height of the sector is 2.12 cm.

Area of the Sector :

Given :

• Radius = 8.94

• Slant height = 2.12 cm

Formula :

$\sf{\underline{\boxed{A_{Sector} = \dfrac{\it{l}r}{2}}}}$

Substituting the values in it ,we get :

$\sf{\Rightarrow A_{Sector} = \dfrac{2.12 \times 8.94}{2}}$

$\sf{\Rightarrow A_{Sector} = \dfrac{18.95}{2}}$

$\sf{\Rightarrow A_{Sector} = \dfrac{\cancel{18.95}}{\cancel{2}}}$

$\sf{\Rightarrow A_{Sector} = 9.5 cm^{2}}$

Hence ,the area of the Sector is 9.5 cm².

Additional information :

• Surface Area of a Cylinder = 2πr(h + r)

• Curved surface area of a Cylinder = 2πrh

• Surface area of a Cube = 6(a)²

• Curved surface area of a Cube = 4(a)²