Find The Length Of Each Arc Calculator

This Arc Length Calculator helps you find the length of an arc of a circle given the radius and the central angle. Enter the values below to get started.

Calculate Arc Length

Angle (Degrees)	Angle (Radians)	Arc Length
Enter values to see table.

Table showing Arc Length for various angles with the given radius.

What is an Arc Length Calculator?

An Arc Length Calculator is a tool used to determine the distance along the curved line making up an arc of a circle. An arc is a portion of the circumference of a circle. To calculate its length, you need to know the radius of the circle and the central angle subtended by the arc.

This calculator is useful for students, engineers, architects, designers, and anyone working with circular shapes or paths. It simplifies the process of finding the length of a segment of a circle's boundary. Common misconceptions include thinking arc length is the straight-line distance between the arc's endpoints (which is the chord length) or confusing it with the area of the sector.

Arc Length Calculator Formula and Mathematical Explanation

The formula to find the length of an arc (L) depends on whether the central angle (θ) is measured in degrees or radians:

• If the angle θ is in degrees: L = (θ / 360°) * 2 * π * r
• If the angle θ is in radians: L = r * θ

Where:

• L is the arc length
• r is the radius of the circle
• θ is the central angle
• π (pi) is approximately 3.14159

The formula essentially calculates what fraction of the full circle's circumference (2 * π * r) the arc represents, based on the angle θ relative to the full circle's 360° or 2π radians.

Variables Table:

Variable	Meaning	Unit	Typical Range
L	Arc Length	Same as radius (e.g., cm, m, inches)	0 to 2πr
r	Radius	Length units (e.g., cm, m, inches)	> 0
θ	Central Angle	Degrees (°) or Radians (rad)	0-360° or 0-2π rad (can be larger)
π	Pi	Constant	~3.14159

Practical Examples (Real-World Use Cases)

Example 1: Pizza Slice

Imagine a circular pizza with a radius of 18 cm. You cut a slice with a central angle of 45°. What is the length of the crust of that slice (the arc length)?

• Radius (r) = 18 cm
• Angle (θ) = 45°
• Using the formula: L = (45 / 360) * 2 * π * 18 = (1/8) * 36 * π = 4.5 * π ≈ 14.14 cm

The crust length of the pizza slice is approximately 14.14 cm.

Example 2: Curved Path

A garden path is designed as a circular arc with a radius of 10 meters and a central angle of 1.2 radians. How long is the path?

• Radius (r) = 10 m
• Angle (θ) = 1.2 rad
• Using the formula: L = r * θ = 10 * 1.2 = 12 meters

The length of the curved garden path is 12 meters.

How to Use This Arc Length Calculator

Our Arc Length Calculator is simple to use:

1. Enter the Radius (r): Input the radius of the circle to which the arc belongs.
2. Enter the Central Angle (θ): Input the angle subtended by the arc at the center of the circle.
3. Select Angle Unit: Choose whether the angle you entered is in degrees or radians from the dropdown menu.
4. Calculate: The calculator will automatically update the arc length and other details as you type or when you click "Calculate".
5. Read Results: The primary result is the arc length. You'll also see intermediate values like the angle in radians (if you entered degrees) and the full circumference. The formula used is also displayed.
6. View Chart and Table: The chart and table dynamically update to show the relationship between angle and arc length for the given radius.

The Arc Length Calculator gives you the direct measurement of the curve.

Key Factors That Affect Arc Length Calculator Results

• Radius (r): The arc length is directly proportional to the radius. A larger radius results in a longer arc for the same central angle.
• Central Angle (θ): The arc length is also directly proportional to the central angle. A larger angle covers more of the circumference, leading to a longer arc.
• Angle Unit: Using the correct formula based on whether the angle is in degrees or radians is crucial. Our Arc Length Calculator handles the conversion if needed.
• Precision of π: The value of π used in the calculation affects precision. We use a high-precision value of `Math.PI`.
• Measurement Accuracy: The accuracy of the input radius and angle values directly impacts the accuracy of the calculated arc length.
• Full Circle vs. Arc: The arc length is always a fraction (or multiple, if angle > 360°) of the full circle's circumference (2πr).

Frequently Asked Questions (FAQ)

What is the difference between arc length and chord length?

Arc length is the distance along the curve of the arc, while chord length is the straight-line distance between the two endpoints of the arc.

Can the central angle be greater than 360 degrees or 2π radians?

Yes, an angle can be greater than 360° or 2π rad, representing more than one full rotation. The Arc Length Calculator will still calculate the corresponding arc length, which would be longer than the circumference.

What if the radius is zero?

If the radius is zero, the circle is just a point, and the arc length will be zero regardless of the angle.

What if the angle is zero?

If the angle is zero, the arc length is zero.

How do I convert degrees to radians?

To convert degrees to radians, multiply the angle in degrees by π/180. Our Arc Length Calculator does this automatically if you input the angle in degrees.

How do I convert radians to degrees?

To convert radians to degrees, multiply the angle in radians by 180/π.

In what units is the arc length measured?

The arc length is measured in the same units as the radius (e.g., meters, centimeters, inches).

Is the Arc Length Calculator free to use?

Yes, this Arc Length Calculator is completely free to use.

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