Question

The central angles of two sectors of circles of radii 7 cm and 21 cm are respectively 120° and 40°. Find the areas of the two sectors as well as the lengths of the corresponding arcs. What do you observe?

Answer 1

Let the lengths of the corresponding arc be l₁ and l₂.

Radius of the sector PO₁QP = 7 cm

Radius of the sector AO₂BA = 21 cm

And, Central angle of sector PO₁QP = 120°

Central angle of sector AO₂BA = 40°

Area of sector with angle `O_1 = (pir^2)/360 xx 120`

= `(pi7^2)/360 xx 120`

= `154/3  cm^2`

Area of sector with angle `O_2 = (pir^2)/360 xx 40`

= `(pi(21)^2)/360 xx 40`

= 154 cm²

Arc length of sector PO₁QP = Central angle × radius

= `120 xx 7 xx pi/180`

= `44/3` cm

Arc length of sector AO₂BA = Central angle × radius

= `40 xx 21 xx pi/180`

= `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.