Arc Length is defined as the distance between the two points placed on the circle's circumference and measured along the circumference. Arc length is the curved distance along the circumference of the circle. The length of the arc between two points is always greater than the chord between those two points.
Arc length of a circle can be calculated with the radius and central angle using the arc length formula, L = θ × r, where θ is in radians. Length of an Arc = θ × (π/180) × r, where θ is in degree.
In this article, we will learn about, Arc Length, Arc length Formula, Arc length Examples, and others in detail.
What is Arc Length?
Arc length is defined as the circular distance between two points along the circumference of a 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.
Arc Length Definition
Arc Length is defined as the length of any curve and din general length of the arc is generally calculated for the circle.
The figure given below shows the arc AB when the radius is r and the central angle is θ.
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)
where,
- r is the Radius of Circle
- θ is the Angle in Degrees
- L is the Arc Length
Arc length when the angle is represented in radians
1 radian = π/180°
Substituting the value of radian in first formula
L = 2πr × (θ × / 360)
L = r θ
where,
- r is the Radius of 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 Circle
- θ is the Angle in Degrees
We 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 a 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 Arc with Angle π Radians,
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: Note the central angle and length of the radius of the given arc.
Step 2: Use the arc length formula given above according to the value of the angle in degrees or radians.
Step 3: Simplify the above equation to get the required answer.
How to Find Arc Length of a Curve
The arc length of a circle's arc can be determined using various methods and formulas depending on the available data. Here are some key scenarios:
- Calculating arc length with the radius and central angle
- Calculating arc length without the radius
- Calculating arc length without the central angle
Area of Sector of a Circle
Area of Sector of a Circle a circle is calculated using the sector of a circle formula. The sector of a circle formula is,
A = r2×(θ/2)
For example, Find the sector of area with central angle π/6 and radius 10 cm.
Given,
A = r2×(θ/2)
A = (10)2(π/6)×(1/2) = 25/3π
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Examples on Arc Length
Example 1: Find the length of the arc with a radius of 2 m and angle π/2 radians.
Solution:
Formula to calculate the length of the arc(L) is, L = r θ
Given:
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:
Formula to calculate the arc length for the function is, L = ∫√(1 + (dy/dx)2)dx
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 = x + c
Limit of integral is [2, 4]
L = 4 - 2 = 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:
Formula to Calculate the Length of the Arc(L) is, L = 2πr × (θ / 360)
Given:
Length of Arc = 2πr × (60 / 360) = 2πr × 1/6 = 2 × 3.1415 × 5/6 (π = 3.1415)
Length of Arc = 5.235 cm
Thus, the length of the arc is 5.235 cm
Example 4: Find the length of the arc with a radius of 0.5m and an angle of π/4 radians.
Solution:
Formula to calculate the length of the arc(L) is, L = r θ
Given:
Length of arc = 0.5 × π/4 = 0.392 m
Thus, the length of the arc is 0.392 m
Example 5: Find the length of the arc with a radius of 10 cm and an angle of 135°.
Solution:
Formula to calculate the length of the arc(L) is, L = 2πr × (θ / 360)
Given:
Length of Arc = 2πr × (135/360) = (2 × 3.1415 × 10 × 135)/360°
Length of Arc = 23.56 cm
Thus, the length of the arc is 23.56 cm.
Example 6: Find the length of the arc with a radius of 20 mm and angle π/6 radians.
Solution:
Formula to calculate the length of the arc is, L = r θ
Given:
Length of Arc = 20 × π/6 = 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:
Formula to calculate the length of the arc(L) is, L = 2πr × (θ / 360)
Given:
Length of Arc = 2πr × (90 / 360) = 2πr × 1/4 = 2 ×3.1415 × 2 × 1/4 = 3.1415 cm
Thus, the length of the arc is 3.1415 cm.
Practice Questions Based on Arc Length
Q1. Find the Length of Arc with a radius of 12 cm and an Angle of 60°.
Q2. Find the Radius of Arc with Length of Arc with 26 cm and Angle 60°.
Q3. Find the Length of Arc with a radius of 9 cm and an Angle of 45°.
Q4. Find the Radius of Arc with Length of Arc with 6 cm and Angle 30°.
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