Formula to find perimeter of the sector is
= l + 2r
where 'l' is the length of the minor arc AB.
Formula to find length of the arc is
l = θ/360° ⋅ 2∏r
Formula to find area of sector is
A = θ/360° ⋅ ∏r2 square units
or
A = rl / 2 square units
Example 1 :
Find the perimeter of the sector PQR shown below.
Solution :
Length of the arc is
l = θ/360° ⋅ 2∏r
Substitute θ = 60°, r = 42 and ∏ = 22/7.
l = 60°/360° ⋅ 2 ⋅ 22/7 ⋅ 42
l = 1/6 ⋅ 2 ⋅ 22 ⋅ 6
l = 1 ⋅ 2 ⋅ 22
l = 44 cm
Perimeter of sector is
= l + 2r
Substitute l = 44 and r = 42.
= 44 + 2(42)
= 44 + 84
= 128 cm
So, length of the arc is 128 cm.
Example 2 :
Find the perimeter of the sector AOB shown below.
Solution :
Length of the arc is
l = θ/360° ⋅ 2∏r
Substitute θ = 120°, r = 21 and ∏ = 22/7.
l = 120°/360° ⋅ 2 ⋅ 22/7 ⋅ 21
l = 1/3 ⋅ 2 ⋅ 22 ⋅ 3
l = 1 ⋅ 2 ⋅ 22
l = 44 cm
Perimeter of sector is
= l + 2r
Substitute l = 44 and r = 21.
= 44 + 2(21)
= 44 + 42
= 86 cm
So, length of the arc is 86 cm.
Example 3 :
Find the perimeter of sector whose area is 324 square cm and the radius is 27 cm.
Solution :
Area of sector = 324 cm2
rl / 2 = 324
Multiply each side by 2.
rl = 648
Substitute 27 for r.
27l = 648
Divide each side by 27.
l = 24 cm
Perimeter of sector is
= l + 2r
= 24 + 2(27)
= 24 + 54
= 78 cm
Example 4 :
Find the length of arc, if the perimeter of sector is 45 cm and radius is 10 cm.
Solution :
Perimeter of sector = 45 cm
l + 2r = 45
Substitute 10 for r.
l + 2 (10) = 45
l + 20 = 45
l = 45 - 10
l = 35 cm
Example 5 :
Find the radius of sector whose perimeter of the sector is 30 cm and length of the arc is 16 cm.
Solution :
Perimeter of sector = 30 cm
l + 2r = 30
Substitute 16 for l.
16 + 2r = 30
Subtract 16 from each side.
2r = 14
Divide each side by 2.
r = 7 cm
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