Question

A hand fan is made up of cloth fixed in between the metallic wires. It is in the shape of a sector of a circle of radius 21 cm and of angle 120° as shown in the figure. Calculate the area of the cloth used and also find the total length of the metallic wire required to make such a fan.​

Answer 1

Given :-

• fan is in the shape of a sector of circle of radius 21 cm
• and an angle of 120°

To find :-

• area of cloth used to make fan
• total length of metallic wire required to make fan

Figure :-

$\setlength{\unitlength}{1cm}\thicklines\begin{picture}(6,4)\put(0,0){\line(1,1){2}}\put(0,0){\line(-1,1){2}}\qbezier(-2,2)(0,3)(2,2)\qbezier(-0.5,0.5)(0,0.7)(0.5,0.5)\put(-0.4,1){ 120^{\circ} }\put(-2.5,2.2){ A }\put(2,2.2){ B }\put(-0.2,-0.5){ O }\put(1.4,1){ 21\;cm }\end{picture}$

→ AB is the arc length.

→ θ = 120°

→ Radius = OB = 21 cm

Knowledge Required :-

▶ Formula for calculating area of sector

$\boxed{\sf{area\;of\;sector=\frac{\pi r^2\times \theta}{360}}}$

(where r is the radius of sector and θ is the angle of the sector )

▶ Formula for calculating arc length

$\boxed{\sf{arclength=\frac{2\pi r \times \theta}{360}}}$

(where r is the radius of sector , θ is the angle of subtended by arc at centre )

Solution :-

Calculating area of cloth required for making Fan

$\implies\sf{area\;of\;cloth\;required=\frac{\pi r^2 \times \theta}{360}}\\\\\implies\sf{ar(cloth\;required\;for\;fan)=\frac{\frac{22}{7}\times(21)^2\times 120}{360}}\\\\\implies\boxed{\sf{ar(cloth\;required\;for\;fan)=462\;\;cm^2}}$

Calculating length of metallic wire required for fan

$\implies\sf{metallic\;wire\;required=arclength+2(radius)}\\\\\implies\sf{metallic\;wire\;required=\frac{2\pi r\times \theta}{360}\;+ 2 (21) }\\\\\implies\sf{mettalic \;wire\;required=\frac{2\times\frac{22}{7}\times 21 \times 120 }{360}+42}\\\\\implies\sf{metallic\;wire\;required=44 + 42 }\\\\\implies\boxed{\sf{metallic\;wire\;required=86\;\;cm}}$