Table of Content

• What is Arc Length?
• Arc Length Formula
• How to Find Arc Length?
• Area of Sector of a Circle
• Examples on Arc Length
• Practice Questions Based on Arc Length

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,

• r = 10 cm
• θ = π/6

A = r²×(θ/2)

A = (10)²(π/6)×(1/2) = 25/3π

People Also View:

• Equation of a Circle
• Degrees To Radians
• Radians to Degrees

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:

• r = 2 m
• 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: 

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:

• r = 5 cm
• θ = 60°

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:

• r = 0.5 m
• θ = π/4 Radians

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:

• r = 10 cm
• θ = 135°

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:

• r = 20 mm
• θ = π/6 Radians

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:

• r = 2 cm
• θ = 90°

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°.

FAQs on Arc Length

What is Arc Length of a Circle?

Arc length of a circle is the length made by the arc which is measured along its circimference.

What is Unit for Length of Arc?

Length of arc is of a circle is either measured in m or in cm.

Learn about, cm to m conversion

Why is Arc Length Measured in Radians?

Angles are measured in radians and arc length is a measurement of distance, thus it cannot be measured in radians.

How to Find the Circumference if the Arc Length (l) and Central Angle (θ) are given?

When arc length (l) and central angle (θ) is given then the circumference is given by the formula, Circumference = (360°/θ)×(l).

How do You Calculate Arc Length?

Arc Length is calculated using the Arc Length Formula that is, L = 2πr × (θ / 360°).

Is Arc Length and Circumference the Same?

No, Arc Length and Circumference are not the same.

What is Area of a Circle?

Area of a Circle is the area inclosed inside the circumference of the circle.

What is Arc Length Equation?

Arc Length Equation is used to find the Arc Length of any curve, and the arc length equation is,

• Arc Length = θ × r, where θ is in Radian
• Arc Length = rθ × (π/180), where θ is in Degree