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 arcs be l₁ and l₂.

Given that, radius of sector PO₁QP = 7 cm

And radius of sector AO₂BA = 21 cm

Central angle of the sector PO₁QP (θ₁) = 120°

And central angle of the sector AO₂BA (θ₂) = 40°

∴ Area of the sector with central angle O₁

= `(pi"r"^2)/360^circ xx θ_1`

= `(pi(7)^2)/360^circ xx 120^circ`

= `22/7 xx (7 xx 7)/360^circ xx 120^circ`

= `(22 xx 7)/3`

= `154/3 "cm"^2`

And area of the sector with central angle O₂

= `(pi"r"^2)/360^circ xx θ_2`

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

= `22/7 xx (21 xx 21)/360^circ xx 40^circ`

= `(22 xx 3 xx 21)/9`

= 22 × 7

= 154 cm²

Now, corresponding arc length of the sector PO₁QP

= `θ_1/360^circ xx 2pi"r"`

= `120^circ/360^circ xx 2 xx 22/7 xx 7`

= `44/3 "cm"` 

And corresponding arc length of the sector AO₂BA

= `θ_2/360^circ xx 2pi"r"`

= `40^circ/360^circ xx 2 xx 22/7 xx 21`

= `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.