Arc length is a fundamental concept in geometry and calculus. It helps us measure the distance along a curved line instead of a straight one. From circular paths to complex curves, arc length plays an important role in mathematics, physics, and engineering.
In this guide, you will learn what arc length is, how to calculate it using formulas, when to use integrals, and how to understand arc length using simple charts and examples.
What Is an Arc?
An arc is a portion of the circumference of a circle or any curved path between two points.
- A circle arc is part of a circle’s boundary.
- A curve arc refers to a segment of any smooth curve.
The length of an arc is the distance measured along the curved path, not the straight-line distance between the endpoints.
What Is Arc Length?
Arc length is the measure of the distance along a curved line.
For straight lines, distance is easy to calculate. But for curves, we use special formulas and, in advanced cases, calculus-based integrals.
Arc Length of a Circle
The arc length of a circle depends on:
- The radius (r) of the circle
- The central angle (θ) subtended by the arc
Arc Length Formula (Radians)
When the angle is measured in radians, the arc length formula is:
Arc Length (L) = r × θ
Where:
- r = radius of the circle
- θ = central angle in radians
This is the most important and commonly used arc length formula.
Arc Length Formula (Degrees)
If the angle is given in degrees, use this formula:
L = (θ ÷ 360) × 2πr
Where:
- θ = angle in degrees
- 2πr = circumference of the circle
How to Calculate Arc Length (Step-by-Step)
Example 1: Arc Length Using Radians
A circle has a radius of 10 cm and a central angle of π/3 radians.
Solution:
L = r × θ
L = 10 × π/3
L ≈ 10.47 cm
Example 2: Arc Length Using Degrees
A circle has:
- Radius = 14 m
- Angle = 90°
Solution:
L = (90 ÷ 360) × 2π × 14
L = 7π
L ≈ 21.99 m
Arc Length of a Curve (Calculus)
For non-circular curves, we use arc length integrals.
Arc Length Formula (Calculus)
For a function y = f(x) from x = a to x = b, the arc length formula is:
L = ∫ from a to b √(1 + (dy/dx)²) dx
This formula calculates the length of a smooth curve using integration.
Example: Arc Length Using Integration
Find the arc length of the curve:
y = x² from x = 0 to x = 1
Steps:
- dy/dx = 2x
- Square it: (2x)² = 4x²
- Add 1: 1 + 4x²
- Take square root: √(1 + 4x²)
- Integrate from 0 to 1
This method is commonly used in calculus and engineering.
Length of a Curve Formula (Parametric Form)
For parametric equations:
- x = f(t)
- y = g(t)
The arc length formula is:
L = ∫ √[(dx/dt)² + (dy/dt)²] dt
This is useful for complex paths such as spirals and cycloids.
Area of an Arc (Sector of a Circle)
The area of a sector is related to arc length.
Sector Area Formula (Radians)
Area (A) = (r² × θ) ÷ 2
Where:
- r = radius
- θ = angle in radians
Example: Sector Area
Radius = 12 cm
Angle = π/6
A = (12² × π/6) ÷ 2
A = 12π cm²
Arc Length Chart (Quick Reference)
| Angle (Degrees) | Angle (Radians) | Arc Length Formula |
| 30° | π/6 | L = rπ/6 |
| 45° | π/4 | L = rπ/4 |
| 60° | π/3 | L = rπ/3 |
| 90° | π/2 | L = rπ/2 |
| 180° | π | L = rπ |
| 360° | 2π | L = 2πr |
How to Find Arc Length Without Radius
If the sector area and angle are known:
- Multiply the area by 2
- Divide by the angle (in radians)
- Take the square root to find the radius
- Multiply radius by the angle
This method is useful in reverse calculations.
Arc Length Calculator (When to Use)
An arc length calculator is useful when:
- Values are large
- You need fast results
- You want to avoid manual integration
However, understanding the formula is essential for exams and problem-solving.
Common Mistakes When Calculating Arc Length
- Using degrees instead of radians in L = rθ
- Forgetting to convert angles
- Confusing chord length with arc length
- Applying circle formulas to non-circular curves
Frequently Asked Questions (FAQs)
What is the arc length of a circle?
It is the distance measured along a curved portion of the circle’s circumference.
Does arc length require radians?
Radians are not mandatory, but they make calculations much simpler.
Is arc length the same as circumference?
No. Circumference is the total length of a circle. Arc length is only a part of it.
What is the arc length formula in calculus?
L = ∫ √(1 + (dy/dx)²) dx
Final Summary
Arc length helps measure curved distances accurately.
- For circles, use L = rθ
- For degrees, use (θ ÷ 360) × 2πr
- For curves, use integral-based formulas
Understanding arc length builds a strong foundation for geometry, calculus, and real-world applications.