In the diagram, the area of minor sector XOY is shaded. θ is measured in radians. We use a ratio to obtain :
area of sector/area of circle = θ/2π
A/πr^{2} = θ/2π
A = (1/2)θr^{2}
For θ in radians, area of sector A = (1/2)θr^{2}
For θ in degrees, area of sector A = (θ/360) × πr^{2}
Problem 1 :
Use radians to find the arc length and area of a sector of a circle of :
a. radius 9 cm and angle 7π/4
b. radius 4.93 cm and angle 4.67 radians.
Solution :
a) radius r = 9 cm
angle θ = 7π/4
arc length = rθ
= 9 × 7π/4
= 9 × 7(3.14)/4
= 197.82/4
= 49.455 cm
Area of sector = 1/2(θr^{2})
= 1/2 × 7π/4 × (9)^{2}
= 1/2 × 7(3.14)/4 × 81
= 1780.38/8
= 222.55 cm^{2}
b) radius r = 4.93 cm
angle θ = 4.67
arc length = rθ
= 4.93 × 4.67
= 23.02 cm
Area of sector = 1/2(θr^{2})
= 1/2 × 4.67× (4.93)^{2}
= 1/2 × 4.67 × 24.30
= 56.74 cm^{2}
Problem 2 :
A sector has an angle of 1.19 radians and an area of 20.8 cm^{2}. Find its :
a. radius b. perimeter
Solution :
a.
Angle θ = 1.19
Area A = 20.8 cm^{2}
area of sector A = (1/2)θr^{2}
20.8 = (1/2) × 1.19 × r^{2}
20.8 = 0.595 × r^{2}
r^{2} = 20.8/0.595
r^{2} = 34.9580
r = 5.91 cm
b.
P = 2r + l
l = θr
l = 1.19 × 5.91
l = 7.03
P = 2(5.91) + 7.03
P = 11.82 + 7.03
p = 18.9 cm
Problem 3 :
Find, in radians, the angle of a sector of :
a. radius 4.3 m and arc length 2.95 m
b. radius 10 cm and area 30 cm^{2}
Solution :
a.
radius r = 4.3 m
arc length l = 2.95 m
Using arc length formula,
l = θr
2.95 = θ × 4.3
θ = 2.95/4.3
θ = 0.686
So, the angle of a sector measures is 0.686 m.
b.
Given, radius r = 10 cm
area A = 30 cm^{2}
area of sector A = (1/2)θr^{2}
30 = (1/2) × θ × (10)^{2}
30 = (1/2) × θ × 100
30 = 50 × θ
θ = 30/50
θ = 0.6
So, the angle of the sector measures is 0.6 cm^{2}.
Problem 4 :
Find θ (in radians ) for each of the following, and hence find the area of each figure :
a.
b.
c.
Solution :
a. By observing the figure,
radius r = 8 cm
area of sector s = 6 cm
s = rθ
6 = 8 × θ
6/8 = θ
θ = 0.75 cm
area = 1/2 × θr^{2}
= 1/2 × 0.75 × 8^{2 }
= 1/2 × 0.75 × 64
area = 24 cm^{2}
b. By observing the figure,
radius r = 5 cm
area of sector s = 8.4 cm
s = rθ
8.4 = 5 × θ
8.4/5 = θ
θ = 1.68 cm
area = 1/2 × θr^{2}
= 1/2 × 1.68 × 5^{2}
= 1/2 × 1.68 × 25
area = 21 cm^{2}
c. By observing the figure,
radius r = 8 cm
area of sector s = 31.7 cm
s = rθ
31.7= 8 × θ
31.7/8 = θ
θ = 3.9625 cm
area = 1/2 × θr^{2}
= 1/2 × 3.9625 × 8^{2 }
= 1/2 × 3.9625 × 64
area = 126.8cm^{2}
Problem 5 :
Find the arc length and area of a sector of radius 5 cm and angle 2 radians.
Solution :
radius r = 5 cm
angle θ = 2 radians
l = θr
l = 2 × 5
l = 10 cm
So, arc length is 10 cm.
area of sector A = (1/2)θr^{2}
= (1/2) × 2 × (5)^{2}
= 25 cm^{2}
So, area of a sector is 25 cm^{2}.
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM