Arc Length Calculator – Find Arc Length from Radius and Angle

The concept of arc length plays a key role in geometry, trigonometry, and physics describing the distance along the curved edge of a circle or any circular segment. Whether you’re measuring the perimeter of an arc, analyzing motion along a curve, or studying circular geometry, our Arc Length Calculator on Digital Calculator helps you find the precise arc length in seconds.

Simply enter the radius and central angle (in degrees or radians), and our calculator instantly computes the arc length with step-by-step clarity.

What Is Arc Length?

The arc length of a circle is the distance measured along the circle’s circumference between two points. It represents a portion of the circle’s total circumference, corresponding to a given central angle.

Imagine a slice of pizza the crust edge represents the arc, and its length depends on the angle of the slice and the radius of the pizza.

In mathematical terms, the arc length is proportional to the circle’s radius and the angle subtended at the center.

Arc Length Formula

The formula for the arc length (s) depends on whether the angle is measured in degrees or radians.

1. Arc Length Formula (Radians)

s = r × θ
Where:

• s = arc length
• r = radius of the circle
• θ = central angle (in radians)

2. Arc Length Formula (Degrees)

s = (θ / 360) × 2πr
Where:

• θ = central angle (in degrees)
• r = radius
• π ≈ 3.141592653589793

Understanding the Formula

Since a full circle measures 2π radians (or 360°) and its circumference is 2πr, the arc length formula simply takes the proportion of the angle compared to a full circle and multiplies it by the circumference.

That’s why:

• A 180° angle covers half the circle → arc length = πr
• A 90° angle covers a quarter of the circle → arc length = (πr) / 2

How to Use the Arc Length Calculator

The Digital Calculator Arc Length Calculator makes finding arc length simple and accurate:

1. Enter the circle’s radius.
   Example: 10 cm
2. Enter the central angle.
   Example: 60° or π/3 radians
3. Select the angle unit (degrees or radians).
4. Click “Calculate.”
5. The calculator instantly displays:
   • Arc Length (s)
   • Step-by-step calculation
   • Optional circumference and area of the sector

Example Output:
Arc Length = 10.472 cm

Example 1: Using Degrees

Find the arc length for a circle with a radius of 8 cm and a central angle of 60°.

Formula:
s = (θ / 360) × 2πr
s = (60 / 360) × 2 × π × 8
s = (1/6) × 16π
s = 8.3776 cm

Result: Arc length = 8.38 cm

Example 2: Using Radians

Find the arc length for a circle with radius 5 m and angle 2 radians.

Formula:
s = r × θ
s = 5 × 2
s = 10 m

Result: Arc length = 10 m

Example 3: Quarter Circle

Radius = 14 cm, Angle = 90°

s = (90 / 360) × 2πr
s = (1/4) × 2π × 14
s = (1/2) × π × 14 = 22 cm

Result: Arc length = 22 cm

Example 4: Full Circle

Radius = 7 cm, Angle = 360°

s = (360 / 360) × 2πr = 2π × 7 = 43.98 cm

Result: Arc length = 43.98 cm (Full Circumference)

Arc Length, Sector Area, and Circle Relationship

The arc length is closely related to the sector area and circle’s circumference.

This relationship makes the Arc Length Calculator a versatile tool for students and engineers alike.

Arc Length Conversion Table

If the radius increases, multiply these values by r to find the actual arc length.

Arc Length in Real Life

Arc length calculations appear in various disciplines:

1. Engineering and Architecture

Used in the design of arches, curved bridges, domes, and roads.

2. Physics and Astronomy

Used to calculate angular motion, circular trajectories, and orbital paths.

3. Manufacturing and Design

Helpful in CNC machining, gear design, and bending materials.

4. Navigation and Geography

Used to determine distances along circular paths, such as latitudes on Earth.

5. Education

Commonly taught in trigonometry, geometry, and pre-calculus courses.

The Arc Length Calculator makes all these applications practical and error-free.

Deriving Arc Length (Radians Formula)

The radian measure of an angle is defined as the ratio between the arc length and the radius:

θ = s / r

Rearranging,
s = r × θ

This relationship makes radians the most natural unit for circular measurement because it directly relates angle and arc length.

Common Mistakes When Finding Arc Length

1. Using the wrong angle unit:
   Always check if your angle is in degrees or radians before applying the formula.
2. Forgetting to convert degrees to radians:
   To convert degrees to radians:
   θ (radians) = θ (degrees) × (π / 180)
3. Using wrong radius units:
   Ensure consistent units for radius and arc length (e.g., both in cm or meters).
4. Mixing up circumference and arc length:
   The circumference is for 360°, while arc length corresponds to part of that circle.

The Digital Calculator Arc Length Calculator eliminates these errors automatically.

Step-by-Step Conversion Example

Convert a 60° angle to radians, then calculate arc length for r = 10 cm.

Step 1: Convert angle → θ = 60 × (π / 180) = π/3
Step 2: Use s = r × θ
s = 10 × (π/3) = 10.472 cm

 Result: Arc length = 10.47 cm

Arc Length in Terms of Circumference

Since the circumference of a full circle is 2πr, arc length is simply a fraction of it.

s = (θ / 360°) × Circumference
or
s = (θ / 2π) × Circumference (if θ is in radians)

This perspective is helpful for visual learners connecting arc measurement with full circular paths.

Why Use the Arc Length Calculator

• Instant Results: Compute accurate arc lengths in seconds.
• Supports both degrees and radians.
• Step-by-step calculations: Understand every step.
• Educational design: Perfect for students and teachers.
• Accurate and reliable: Uses high-precision π values.
• Completely free and mobile-friendly.

Calculate now with Digital Calculator — geometry made simple.

Why Choose Digital Calculator

At Digital Calculator, we bridge the gap between learning and accuracy. Our tools are built for students, educators, and professionals who value clarity, precision, and simplicity.

From geometry to trigonometry, every calculator on our platform offers:

• Step-by-step explanations
• Accurate computations
• Clean, easy-to-use design

No more manual math errors — just instant, reliable results.

Calculate smarter with Digital Calculator accuracy made simple.