Arc Length Calculator: Find Arc Width From Angle & Radius

Arc Length Calculator

Instantly calculate the width of an arc using the angle and radius of a circle.

Radius (r)

The distance from the center of the circle to its edge.

Radius Unit

Select the unit of measurement for the radius.

Central Angle (θ)

The angle subtended by the arc at the center of the circle.

Angle Unit

Use degrees (full circle is 360°) or radians (full circle is 2π rad).

Arc Length (Width)

Angle in Radians

Full Circumference

The formula used is: Arc Length = Radius × Angle (in radians)

Visual Representation of the Arc

An interactive SVG chart showing the radius and arc based on your inputs.

Example Data Table

Central Angle (Degrees)	Arc Length (for Radius = 10 cm)
30°	5.24 cm
45°	7.85 cm
90°	15.71 cm
180°	31.42 cm
270°	47.12 cm
360°	62.83 cm

This table shows how arc length changes with the angle for a fixed radius.

What is Arc Length?

In geometry, an arc is a segment of the circumference of a circle. Think of a pizza slice; the curved crust at the end is an arc. The “width of an arc” is more formally known as its **arc length**. It’s the distance you would travel if you walked along the curved edge from one endpoint to the other. To accurately **calculate the width of an arc using the angle**, you need two key pieces of information: the circle’s radius and the central angle that forms the arc.

This measurement is distinct from the chord length, which is the straight line connecting the arc’s two endpoints. The arc length will always be longer than the chord length for any arc greater than zero degrees. This concept is fundamental in many fields, including engineering, astronomy, design, and navigation. For a more basic understanding of circles, you might want to look at a Circumference Calculator.

Arc Length Formula and Explanation

The formula to **calculate the width of an arc using the angle** is beautifully simple, but it depends on the unit of the angle. The standard formula requires the angle to be in radians.

s = r × θ

If your angle is in degrees, you must first convert it to radians before using the formula above. The conversion is:

Angle in Radians = Angle in Degrees × (π / 180)

So, the combined formula when starting with degrees is: s = r × (θ_deg × π / 180). Our calculator handles this conversion for you automatically.

Variables Table

Variable	Meaning	Unit (auto-inferred)	Typical Range
s	Arc Length	Length (cm, m, in, ft)	Greater than 0
r	Radius	Length (cm, m, in, ft)	Greater than 0
θ	Central Angle	Radians (rad) or Degrees (°)	Typically 0 to 360° or 0 to 2π rad

Practical Examples

Let’s walk through two examples to see how to calculate the width of an arc using the angle.

Example 1: Using Degrees

Imagine a decorative garden path that forms a circular arc. The circle it belongs to has a radius of 15 feet, and the path covers an angle of 60 degrees.

Inputs: Radius (r) = 15 ft, Angle (θ) = 60°
Step 1: Convert angle to radians. 60° × (π / 180) = π/3 radians ≈ 1.047 rad
Step 2: Calculate arc length. s = 15 ft × 1.047 rad
Result: The arc length (path length) is approximately 15.71 feet.

Example 2: Using Radians

An engineer is designing a curved component. The radius is specified as 50 centimeters, and the central angle is 2.5 radians.

Inputs: Radius (r) = 50 cm, Angle (θ) = 2.5 rad
Step 1: The angle is already in radians. No conversion needed.
Step 2: Calculate arc length. s = 50 cm × 2.5 rad
Result: The arc length of the component is 125 centimeters. To explore how this relates to a full circle, consider using a Circle Area Calculator.

How to Use This Arc Length Calculator

Enter the Radius: Input the radius of the circle in the first field.
Select Radius Unit: Choose the appropriate unit of length for your radius (e.g., meters, inches). The result will be in this unit.
Enter the Central Angle: Input the angle of the arc.
Select Angle Unit: CRITICAL: Specify whether the angle you entered is in **degrees** or **radians**. The calculation depends heavily on this setting.
Interpret the Results: The calculator instantly shows the final arc length. It also provides intermediate values like the angle in radians (if you entered degrees) and the total circumference for context. The visual chart updates in real-time to help you see the arc you’ve defined. To understand angles better, our Angle Conversion Tool can be very helpful.

Key Factors That Affect Arc Length

Radius (r): This is a directly proportional relationship. If you double the radius while keeping the angle constant, you double the arc length. A larger circle will have a longer arc for the same angular segment.
Central Angle (θ): This is also a directly proportional relationship. For a fixed radius, doubling the central angle will double the arc length.
Angle Units: Failing to use the correct angle unit (degrees vs. radians) is the most common source of error. The direct formula `s = r * θ` only works for radians.
Measurement Precision: The accuracy of your input values for radius and angle directly impacts the accuracy of the result.
Full Rotations: An angle greater than 360° (or 2π radians) means the arc wraps around the circle more than once. Our calculator correctly handles this. A related tool is the Sector Area Calculator.
Application Context: In fields like geography, calculating the distance between two points on Earth (a very large sphere) uses a similar principle, but with spherical geometry.

Frequently Asked Questions (FAQ)

1. What is the difference between arc length and chord length?: Arc length is the distance along the curved path of the circle’s edge. Chord length is the straight-line distance between the arc’s two endpoints. The arc length is always longer.
2. Why do I need to use radians for the formula?: The radian unit is naturally derived from the circle itself, where one radian is the angle created when the arc length equals the radius. This creates the simple `s = r * θ` relationship. Degrees are a more arbitrary human invention (360 in a circle), which requires a conversion factor.
3. Can I calculate arc length without the angle?: Yes, but you need other information. For example, if you know the radius and the chord length, you can first calculate the angle using trigonometry and then find the arc length.
4. What if my angle is larger than 360 degrees?: The formula still works perfectly. An angle of 720°, for instance, means the arc wraps around the circle exactly twice. The arc length would be two times the circumference.
5. How do I calculate the width of an arc using the angle in the real world?: Measure the radius of the circular object. Then, from the center point, measure the angle formed by the start and end points of the arc. Plug these values into the calculator or formula. This is used in construction, landscaping, and manufacturing.
6. Does the unit of the radius matter?: Yes, the unit you use for the radius (e.g., inches, meters) will be the unit of your final arc length. Our calculator lets you select the unit and labels the result accordingly.
7. Is ‘arc width’ the same as ‘arc length’?: Yes, in the context of this calculation, the terms are used interchangeably. “Arc length” is the more formal mathematical term.
8. Can I find the angle if I know the arc length and radius?: Yes, you can rearrange the formula: θ = s / r. This will give you the angle in radians. To convert it to degrees, you can use our Radians to Degrees Calculator.

Related Tools and Internal Resources

If you found this tool useful, you might also be interested in our other geometry and math calculators:

Sector Area Calculator – Calculate the area of a ‘slice’ of a circle.
Circumference Calculator – Find the total distance around a circle.
Circle Area Calculator – Calculate the total area enclosed by a circle.
Radians to Degrees Calculator – Convert between angle units.
Angle Conversion Tool – A general tool for various angle conversions.
Pythagorean Theorem Calculator – Useful for related right-triangle calculations.