Arc Length Calculator in Degrees

Calculating arc length in degrees is essential for geometry, engineering, and physics applications. This calculator provides an accurate way to determine the length of an arc when you know the radius and central angle in degrees.

What is Arc Length?

Arc length is the distance along a curve of a circle. It's a fundamental concept in geometry and trigonometry. When dealing with circular objects or systems, knowing the arc length helps in various practical applications such as:

Designing circular paths and tracks
Calculating the length of cables wrapped around pulleys
Determining the circumference of circular objects
Analyzing the motion of objects in circular paths

The arc length depends on two main factors: the radius of the circle and the central angle that subtends the arc. When the angle is given in degrees, we need to convert it to radians for calculations since most trigonometric functions use radians.

Arc Length Formula

The formula to calculate arc length (L) when the angle is in degrees is:

L = (θ/360) × 2πr

Where:

L = Arc length
θ = Central angle in degrees
r = Radius of the circle
π (pi) ≈ 3.141592653589793

This formula works because a full circle (360 degrees) has a circumference of 2πr. The fraction θ/360 gives the proportion of the full circle that the arc represents.

Note: For small angles, the arc length is approximately equal to the radius times the angle in radians (L ≈ rθ when θ is in radians). However, for angles greater than about 5 degrees, the exact formula should be used.

How to Use the Calculator

Using our arc length calculator is simple:

Enter the radius of the circle in the first input field
Enter the central angle in degrees in the second input field
Click the "Calculate" button
The calculator will display the arc length in the result box
Optionally, you can view a visualization of the arc

The calculator automatically handles unit conversions and provides clear results. You can also reset the form to start a new calculation.

Worked Examples

Example 1: Small Angle

Suppose you have a circle with radius 5 meters and a central angle of 30 degrees. Calculate the arc length.

Using the formula:

L = (30/360) × 2π × 5 = (1/12) × 10π ≈ 2.61799 meters

Example 2: Large Angle

For a circle with radius 10 centimeters and a central angle of 120 degrees, calculate the arc length.

Using the formula:

L = (120/360) × 2π × 10 = (1/3) × 20π ≈ 20.94395 centimeters

Example 3: Full Circle

What is the arc length when the angle is 360 degrees (full circle) with radius 7 meters?

Using the formula:

L = (360/360) × 2π × 7 = 1 × 14π ≈ 43.98229 meters

This should equal the circumference of the circle (2πr = 14π meters).

FAQ

What is the difference between arc length and chord length?

Arc length is the distance along the curve of the circle, while chord length is the straight-line distance between the two endpoints of the arc. For small angles, these values are approximately equal, but they diverge as the angle increases.

Can I use this calculator for angles greater than 360 degrees?

No, this calculator is designed for angles between 0 and 360 degrees. For angles greater than 360 degrees, you would need to calculate the equivalent angle within one full rotation (using modulo 360) before using this calculator.

Is the arc length always less than the circumference?

Yes, the arc length is always less than or equal to the circumference of the circle. It equals the circumference only when the angle is 360 degrees (a full circle).

How accurate is this calculator?

This calculator uses precise mathematical formulas and JavaScript's built-in trigonometric functions to provide accurate results. The calculations are performed with floating-point arithmetic, which is standard for most scientific calculators.