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?
Given: 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°
To find: Area of two sectors and length of arcs.
Explanation:
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)
Area of first sector = 
= 
= 51.33 cm2
Area of second sector = 
= 
= 154 cm2
Let the lengths of the corresponding arc be l1 and l2.
Now, arc length of first sector = Radius × Central Angle (in radians)
= 
= 44/3 cm
Now, arc length of second sector = Radius × Central Angle (in radians)
= 
= 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.