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Arc Length
Arc length is defined as the distance between the two points placed on the circumference of the circle and measured along the circumference. Arc length is the curved distance along the circumference of the circle. Length of the arc between two points is always greater than the chord between those two points.
What is Arc Length?
The arc length is defined as the circular distance between two points along the circumference of the circle. The length of the arc is directly dependent on the radius and central angle of the circle. The central angle is the angle subtended by the endpoints of the arc to the center of the circle. It is denoted by θ. It is measured both in degrees and radians. The figure given below shows the arc AB when the radius is r and the central angle is θ.
Arc Length Formula
Length of the arc is calculated using different formulas, the formula used is based on the central angle of the arc. Central angle is measured in degrees or radians, and accordingly, the length of an arc of the circle is calculated. For a circle, the formula for arc length formula is θ times the radius of the circle.
| Arc Length Formula (θ in degrees) | s = 2×π×r ×(θ/360°) |
| Arc Length Formula (θ in radians) | s = θ × r |
| Arc Length Formula (Integral Form) | s = ∫√(1 + (dy/dx)2dx |
There are different cases that are used accordingly to find the required Arc Length
Case 1: When Radius and Angle are given
Formula to calculate the length of an arc is given by:
L = 2πr × (θ / 360)… (1)
where
r is the radius of the circle
θ is the angle in degrees
L is the Arc lengthArc length when the angle is represented in radians
1 radian = π/180°
Substituting the value of radian in equation (1)
L = 2πr × (θ × / 360)
L = r θ…(2)
where,
r is the radius of the circle
θ is the angle in radians.
Case 2: When Area and Central Angle of the Arc are given
Formula to calculate the length of an arc is given by:
L = 2πr × (θ / 360)
where,
r is the radius of the circle
θ is the angle in degreesWe need to find the radius of the circle from the given area. After finding the radius, we will substitute the value of radius in the formula.
Area of the circle = πr2
Example: If area of the circle is 314 m2 and centeral angle of the arc is π radian find the length of the arc.
Sloution:
πr2 = 314 m2
r2 = 314/π (π = 3.14)
r2 = 314/3.14
r2 = 100
r = √100 = 10 m
Length of the arc with angle π radians will be:
L = r θ
L = 10 × π
L = 10 × 3.1415
L = 31.415 m
The value of r can be used in the same formula, as discussed above.
Case 3: Arc length In Integral Form
Arc length in integral form is given by:
L = ∫√(1 + (dy/dx)2)dx
where,
Y is the f(x) function
limit of integral is [a, b]
How to Find Arc Length?
Use the steps given below to find the Arc length of the given arc.
Step 1: Mark the central angle and length of the radius of the given arc.
Step 2: Use the formula as given above according to the value of the angle in degrees or radians accordingly.
Step 3: Simplify the above equation to get the required answer.
Also, Check
Solved Examples on Arc Length
Example 1: Find the length of the arc with a radius of 2m and angle π/2 radians.
Solution:
The formula to calculate the length of the arc is given by:
L = r θ
Where,
L is the length of the arc
Given: r = 2m and θ = π/2 radians
Length of arc = 2 × π/2
Length of arc = π
(π = 3.1415)
Length of arc = 3.1415 m
Thus, the length of the arc is 3.1415 m.
Example 2: Find the length of the arc of function f(x) = 8 between x =2 and x = 4.
Solution:
The formula to calculate the arc length for the function is given by:
L = ∫√(1 + (dy/dx)2)dx
The limit of integral is [a, b]
Substituting the values a = 2, b = 4, and y = 6 or dy/dx = 0 in the above formula,
L = ∫√(1 + (0)2)dx
L = ∫√1 dx
L = ∫1 dx
L = x
(Integral of 1 is x)
The limit of integral is [2, 4]
L = (4 – 2)
L = 2
Thus, the length of the arc of function f(x) = 8 between x = 2 and x = 4 is 2.
Example 3: Find the length of the arc with a radius of 5cm and an angle of 60°.
Solution:
The formula to calculate the length of the arc is given by:
L = 2πr × (θ / 360)
Where,
L is the length of the arc
Given: r = 5cm and θ = 60°
Length of arc = 2πr × (60 / 360)
Length of arc = 2πr × 1/6
Length of arc = 2 × 3.1415 × 5/6
(π = 3.1415)
Length of arc = 5.235cm
Thus, the length of the arc is 5.235cm
Example 4: Find the length of the arc with a radius of 0.5m and an angle of π/4 radians.
Solution:
The formula to calculate the length of the arc is given by:
L = r θ
Where,
L is the length of the arc
Given: r = 0.5m and θ = π/4 radians
Length of arc = 0.5 × π/4
Length of arc = 0.392 m
(π = 3.1415)
Thus, the length of the arc is 0.392 m
Example 5: Find the length of the arc with a radius of 10cm and an angle of 135°.
Solution:
The formula to calculate the length of the arc is given by:
L = 2πr × (θ / 360)
Where,
L is the length of the arc
Given: r = 10cm and θ = 135°
Length of arc = 2πr × (135/360)
Length of arc = (2 × 3.1415 × 10 × 135)/360°
(π = 3.1415)
Length of arc = 23.56cm
Thus, the length of the arc is 23.56cm.
Example 6: Find the length of the arc with a radius of 20mm and angle π/6 radians.
Solution:
The formula to calculate the length of the arc is given by:
L = r θ
Where,
L is the length of the arc
Given: r = 20mm and θ = π/6 radians
Length of arc = 20 × π/6
Length of arc = 10.47 mm
(π = 3.1415)
Thus, the length of the arc is 10.47 mm
Example 7: Find the length of the arc with a radius of 2 cm and an angle of 90°.
Solution:
The formula to calculate the length of the arc is given by:
L = 2πr × (θ / 360)
Where,
L is the length of the arc
Given: r = 2cm and θ = 90°
Length of arc = 2πr × (90 / 360)
Length of arc = 2πr × 1/4
Length of arc = 2 ×3.1415 × 2 × 1/4
(π = 3.1415)
Length of arc = 3.1415 cm
Thus, the length of the arc is 3.1415 cm.
FAQs on Arc Length
Question 1: What is the Arc Length of a Circle?
Answer:
Arc length of a circle is the length made by the arc which is measured along its circimference.
Question 2: Length of the arc is measured in which unit?
Answer:
Length of arc is of a circle is either measured in m or in cm.
Question 3: Does arc length is measured in radians?
Answer:
Angles are measured in radians and arc length is a measurement of distance, thus it cannot be measured in radians.
Question 4: How do you find the circumference if the arc length (l) and central angle (θ) are given?
Answer:
When arc length (l) and central angle (θ) is given then the circumference by the formula
Arc Length (L) / Circumference = θ/360º
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