Arc Length Calculator

An arc length calculator computes the distance along the curved edge of a circle — the arc — given the radius and the central angle. This is one of the most fundamental geometric calculations, bridging linear measurement (the radius) with angular measurement (the central angle). The calculator also derives the sector area, chord length, and the arc as a percentage of the full circumference.

The arc length formula, s = r × θ (with θ in radians), is central to geometry, trigonometry, and calculus. It's the reason mathematicians prefer radian measure: in radians, the arc length equals the radius multiplied by the angle — no conversion factor needed. This simplicity makes radian-based formulas cleaner for derivatives and integrals in calculus.

Whether you're calculating the distance a point on a rotating wheel travels, determining the area of a pizza slice, or finding how much fencing is needed for a curved garden path, the arc length calculator handles the math. It supports both degree and radian inputs, converting automatically so you can work in whichever unit is more natural for your problem.

The arc length s is proportional to both the radius r and the central angle θ (in radians). A full circle (θ = 2π) gives s = 2πr — the circumference. Any fraction of that yields a proportionally smaller arc.

The calculator produces four key values alongside a visual SVG diagram of the sector:

The SVG diagram draws the actual sector to scale, showing the radius, the arc, and the chord in a color-coded visual. The angle conversion panel shows the angle in both degrees and radians regardless of input unit.

Three inputs, all straightforward:

1. Enter the radius (r): Type the radius of the circle. Any positive number works — decimals are fine. This is the distance from the center to the circumference.
2. Enter the central angle (θ): Type the angle that the arc subtends at the center. This is the "spread" of the sector — a small angle means a short arc, a large angle approaches the full circle.
3. Select the angle unit: Toggle between Degrees and Radians. The calculator converts internally to radians for computation and displays both units in the results panel. This avoids unit confusion — enter in whichever form your problem uses.
4. Read the outputs: The highlighted result shows arc length as a percentage of circumference. Below it, the four result cards give precise decimal values for each geometric quantity, and the SVG visualization confirms the sector shape.

Arc length calculations are fundamental in engineering and manufacturing. When designing gears, pulleys, or curved tracks, knowing the arc length tells you how much material contacts the surface, which directly determines wear rate and friction. In CNC machining, tool paths along circular arcs require exact arc length values to set feed rates and estimate machining time.

In construction and landscaping, curved walkways, circular flower beds, and arched bridges all need arc length and sector area calculations. A landscape architect planning a curved retaining wall needs the arc length to order the right amount of stone, and the sector area to estimate fill material behind the wall. Road design uses circular arcs for highway curves — the arc length determines the safe stopping sight distance and the correct banking angle.

In education, this calculator helps students visualize the relationship between radius, central angle, arc length, and chord length — concepts that are abstract on a textbook page but become intuitive with interactive feedback. Seeing the SVG diagram update as you change the angle makes the proportional relationship between s and θ immediately clear.