Question

The central angles of two sectors 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

Radius of one sector = r₁ = 7 cm

Radius of second sector = r₂ = 21 cm

Central angle of one sector = 120°

Central angle of second sector = 40°

Central angle of one sector (in radians) = θ₁ = (120π/180)

Central angle of second sector (in radians) = θ₂ = (40π/180)

= 154 cm²

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

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

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

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