find the area of the minor segment of a circle of radius 42cm, if length of the corresponding arc is 44cm.
Answers
Answer:
160.0 cm²
Step-by-step explanation:
R = 42 cm; s = 44 cm
1. Central angle θ (working in radians)
θ = s/R
θ = 44/42
θ = 22/21
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2. Area of sector
A = ½R²θ
A = ½(42)²(22/21)
A = ½(42×21×2)(22/21)
A = 924 cm²
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3. Area of triangle
A = ½R²sinθ
A = ½(42)²sin(22/21)
A = ½ ×1764 × 0.8662
A = 764.0 cm²
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4. Area of segment
Area of segment = Area of sector – area of triangle
= 924 – 764.0
= 160.0 cm²
Answer:
Given :
Radius r = 42 cm
We have :
Length of arc = 44 cm .
i.e. π r Ф / 180 = 44
Ф = 44 × 180 × 7 / 22 × 42
Ф = 2 × 180 / 6
Ф = 2 × 30
Ф = 60 .
Now :
Area of minor segment = Area of sector - Area of corresponding triangle .
Area of minor segment = π r² Ф / 360 - √ 3 / 4 r²
Area of minor segment = r² ( π Ф / 360 - √ 3 / 4 )
Area of minor segment = 42² ( 22 / 7 × 60 / 360 - √ 3 / 4 )
Area of minor segment = 42² ( 22 / 7 × 1 / 6 - √ 3 / 4 )
Area of minor segment = 42² ( 11 / 7 × 1 / 3 - √ 3 / 4 )
Area of minor segment = 42² ( 11 / 21 - √ 3 / 4 )
Area of minor segment = 42² ( 44 - 21 √ 3 / 84 )
Area of minor segment = 42 × 42 ( 44 - 21 √ 3 / 84 )
Area of minor segment = 42 ( 44 - 21 √ 3 / 2 )
Area of minor segment = 21 ( 44 - 21 √ 3 ) cm² .