Arc Length Calculator

An arc length calculator helps you find the distance along a curved line on the perimeter of a circle. Whether you're working on geometry homework, designing curved structures, or calculating circular motion, understanding arc length is essential. This calculator solves for arc length, radius, central angle, sector area, diameter, and chord using standard geometric formulas.

What is Arc Length?

Arc length is the distance measured along the curved line that makes up part of a circle's circumference. It depends on two factors: the radius of the circle and the central angle (the angle formed at the circle's center by the arc's endpoints). The fundamental formula is L = θ × r, where L is arc length, θ is the central angle in radians, and r is the radius. For degrees, the formula becomes L = (θ × π × r) / 180.

The sector area (the pie-shaped region bounded by the arc and two radii) is calculated as A = (1/2) × L × r, or equivalently A = (1/2) × r² × θ. These relationships allow you to solve for any unknown value when you know at least two others.

How to Use the Arc Length Calculator

Enter known values: Input any two values from: central angle, radius, diameter, arc length, or sector area. The calculator needs at least two values to compute the rest.
Select units: Choose your preferred units for angles (degrees, radians) and lengths (meters, feet, etc.). The calculator handles conversions automatically.
View results: All calculated values appear instantly, including arc length, sector area, radius, diameter, and central angle.

For example, if you know the radius is 5 meters and the central angle is 60 degrees, the calculator finds the arc length (5.236 meters), sector area (13.09 square meters), and diameter (10 meters).

Applications of Arc Length

Engineering and construction: Designing curved roads, bridges, arches, and circular building elements requires precise arc measurements.
Manufacturing: Cutting circular parts, bending metal, and creating curved surfaces all involve arc calculations.
Physics: Circular motion problems use arc length to find distance traveled along a curved path.
Navigation: Calculating distances along Earth's surface between two points uses arc length formulas.
Art and design: Creating curved patterns, decorative elements, and circular designs relies on accurate arc measurements.

Formula Variants

The calculator uses these key relationships to solve for different unknowns:

Arc length from angle and radius: L = θ × r (θ in radians)
Radius from arc length and angle: r = L / θ
Central angle from arc length and radius: θ = L / r
Sector area from arc length and radius: A = (1/2) × L × r
Arc length from sector area and radius: L = 2A / r
Diameter from radius: d = 2r

Remember that angles must be in radians for these formulas. The calculator converts between degrees and radians automatically based on your unit selection.

Frequently Asked Questions

What's the difference between arc length and chord length?

Arc length measures the curved distance along the circle's edge, while chord length is the straight-line distance between the arc's endpoints. The arc length is always longer than the chord length for the same section of a circle.

Do I need to use radians or degrees?

You can use either. Select your preferred unit from the dropdown menu, and the calculator handles the conversion. The standard mathematical formula uses radians, but degrees work just as well with the appropriate conversion factor (π/180).

Can the central angle be greater than 360 degrees?

For a single arc on a circle, the central angle should be between 0 and 360 degrees (0 and 2π radians). Larger angles would represent multiple complete rotations, which is uncommon in standard arc calculations.