Formula to find the arc length of a sector is
Example 1 :
Find the length of the arc that is bolded. (Take ∏ ≈ 3.14 and round your answer to one decimal place, if necessary)
Solution :
The formula to find the arc length is
= (Arc Measure / 360°) ⋅ 2Π r
Plug r = 8, Arc Measure = 135° and Π ≈ 3.14
≈ (135° / 360°) ⋅ 2 ⋅ 3.14 ⋅ 8
≈ 18.9
So, the length of the arc is about 18.9 yd.
Example 2 :
Find the length of the arc that is bolded. (Take ∏ ≈ 3.14 and round your answer to one decimal place, if necessary)
Solution :
The formula to find the arc length is
= (Arc Measure / 360°) ⋅ 2Π r
Plug r = 8, Arc Measure = 315° and Π ≈ 3.14
≈ (315° / 360°) ⋅ 2 ⋅ 3.14 ⋅ 8
≈ 44
So, the length of the arc is about 44 cm.
Example 3 :
Find the length of the arc highlighted in red color. (Take ∏ ≈ 3.14 and round your answer to one decimal place, if necessary)
Solution :
Given : Diameter is 4 inches.
Then, the radius is
= Diameter / 2
= 4 / 2
= 2 inches
The formula to find the arc length is
= (Arc Measure / 360°) ⋅ 2Π r
Plug r = 2, Arc Measure = 80° and Π ≈ 3.14
≈ (80° / 360°) ⋅ 2 ⋅ 3.14 ⋅ 2
≈ 2.8
So, the length of the arc is about 2.8 inches.
Example 4 :
In the diagram given below, if QRS is a central angle and m∠QRS = 81°, m∠SRT = 115°, and radius is 5 cm, then find the length of the arc QST. (Take ∏ ≈ 3.14 and round your answer to one decimal place, if necessary)
Solution :
To find the length of the arc QST, first we have to find the arc measure QST or the central angle m∠QRT.
m∠QRT = m∠QRS + m∠SRT
m∠QRT = 81° + 115°
m∠QRT = 196°
The formula to find the arc length is
= (Central Angle / 360°) ⋅ 2Π r
Plug r = 5, Central Angle = 196° and Π ≈ 3.14
≈ (196° / 360°) ⋅ 2 ⋅ 3.14 ⋅ 5
≈ 17.1
So, the length of the arc is about 17.1 cm.
Example 5 :
If m∠LMN = 19° and radius is 15 inches, then find the length of arc LN. (Take ∏ ≈ 3.14 and round your answer to one decimal place, if necessary)
Solution :
To find the length of the arc LN, first we have to find the arc measure LN.
By Inscribed Angle Theorem, we have
1/2 ⋅ Arc Measure = m∠LMN
Multiply both sides by 2.
Arc Measure = 2 ⋅ m∠LMN
Arc Measure = 2 ⋅ 19°
Arc Measure = 38°
The formula to find the arc length is
= (Arc Measure / 360°) ⋅ 2Π r
Plug r = 15, Arc Measure = 38° and Π ≈ 3.14
≈ (38° / 360°) ⋅ 2 ⋅ 3.14 ⋅ 15
≈ 9.9
So, the length of the arc is about 9.9 inches.
Example 6 :
In a circle, if the arc length of Arc AB is 18 cm and the measure of Arc AB is 39°, then find the radius of the circle. (Take ∏ ≈ 3.14 and round your answer to one decimal place, if necessary)
Solution :
Given : The arc length of Arc AB is 18 cm.
So, we have
(Arc Measure / 360°) ⋅ 2Π r = 18
Plug Arc Measure = 39° and Π ≈ 3.14
(39° / 360°) ⋅ 2 ⋅ 3.14 ⋅ r ≈ 18
(39° / 360°) ⋅ 2 ⋅ 3.14 ⋅ r ≈ 18
0.1083 ⋅ 2 ⋅ 3.14 ⋅ r ≈ 18
0.68 ⋅ r ≈ 18
Divide both sides by 0.68.
r ≈ 18 / 0.68
r ≈ 26.5
So, the radius of the circle is about 26.5 cm.
Example 7 :
In a circle, if the arc length of Arc AB is 19 inches and the radius is 29 inches, then find the measure of arc AB. (Take ∏ ≈ 3.14 and round your answer to one decimal place, if necessary)
Solution :
Given : The arc length of Arc AB is 19 inches.
So, we have
(Arc Measure / 360°) ⋅ 2Π r = 19
Plug r = 29 and Π ≈ 3.14
(Arc Measure / 360°) ⋅ 2 ⋅ 3.14 ⋅ 29 ≈ 19
(Arc Measure / 360°) ⋅ 182.12 ≈ 19
(Arc Measure / 360°) ⋅ 182.12 ≈ 19
Multiply both sides by 360° / 182.12
Arc Measure ≈ 19 ⋅ 360° / 182.12
Arc Measure ≈ 19 ⋅ 360° / 182.12
Arc Measure ≈ 37.6°
So, the measure of arc AB is about 37.6°.
Example 8 :
Find the length of the arc highlighted in red color. (Take ∏ ≈ 3.14 and round your answer to one decimal place, if necessary)
Solution :
From the given diagram, we have
m∠MCN + Measure of arc MON = 360°
Plug m∠MCN = 88°
88° + Measure of arc MON = 360°
Subtract 88° from both sides.
Measure of arc MON = 272°
Given : Diameter is 4 inches.
Then, the radius is
= Diameter / 2
= 10 / 2
= 5 ft
The formula to find the arc length is
= (Arc Measure / 360°) ⋅ 2Π r
Plug r = 5, Arc Measure = 272° and Π ≈ 3.14
≈ (272° / 360°) ⋅ 2 ⋅ 3.14 ⋅ 5
≈ 23.7 ft
So, the length of the arc is about 23.7 ft.
Example 9 :
A lasso is made from a rope that is 10 m long. The loop of the lasso has a radius 0.6 m when circular. Find the length of the rope that is not part of the loop.
Solution :
Length of rope which is aligned in a line + circumference of circular loop = 10 m
l + 2Π r = 10
Here r = 0.6 m
l + 2(3.14) (0.6) = 10
l + 3.768 = 10
l = 10 - 3.768
l = 6.232 m
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