Find the area of the shaded region given in figure

#### Solution

Radius of one sector = r_{1} = 7 cm

Radius of second sector = r_{2} = 21 cm

Central angle of one sector = 120°

Central angle of second sector = 40°

Central angle of one sector (in radians) = θ_{1} = `(120π)/180`

Central angle of second sector (in radians) = θ_{2} = `(40π)/180`

Area of first sector = `1/2 r^2 theta_1`

= `1/2 xx 49 xx (120pi)/180`

= `1/2 xx 49 xx (120 xx 22)/(180 xx 7)`

`154/3 = 51.33 cm^2`

Area of second sector = `1/2 r^2 theta_2`

= `1/2 xx 441 xx (400pi)/180`

= `1/2 xx 441 xx (40 xx 22)/(180 xx 7)`

= 154 cm^{2}

Let the lengths of the corresponding arc be l_{1} and l_{2}.

Now, arc length of first sector = Radius × Central Angle (in radians)

= `7 xx (120pi)/80`

= `(7 xx 120 xx 22)/(180 xx 7)`

= `44/3` cm

Now, arc length of second sector = Radius × Central Angle (in radians)

= `21 xx (40pi)/180`

= `(21 xx 40 xx 22)/(180 xx 7)`

= `44/3` cm

Hence, we observe that arc lengths of two sectors of two different circles may be equal but their area need not be equal.