Arc Length Calculator

About This Tool

The Arc Length Calculator is a free tool that computes the arc length, sector area, and chord length of a circular arc given the radius and central angle. It accepts angles in both degrees and radians, automatically converting between the two. Arc length calculations are fundamental in trigonometry, engineering, and physics. Whether you are a student learning about radian measure, an engineer designing curved tracks, or an architect planning circular staircases, this calculator provides instant, accurate results.

The Arc Length Formula

The arc length of a circle is s = rθ, where r is the radius and θ is the central angle measured in radians. This elegantly simple formula is actually the definition of a radian: one radian is the angle subtended by an arc whose length equals the radius. For a full circle, θ = 2π radians, so the arc length is 2πr — the familiar circumference formula. If the angle is given in degrees, convert first using θ_rad = θ_deg × π/180. The linearity of this formula makes it extremely convenient: doubling the angle doubles the arc length, and doubling the radius also doubles the arc length.

Sector Area Formula

The area of a circular sector (the pie-slice region between two radii and the arc) is A = (1/2)r²θ, where θ is in radians. This can be derived by noting that a sector with angle θ is a fraction θ/(2π) of the full circle, so its area is (θ/(2π)) × πr² = (1/2)r²θ. Alternatively, it can be derived using integration in polar coordinates. The sector area formula is essential in probability (calculating areas of circular sectors in pie charts), in mechanical engineering (computing the swept area of rotary valves), and in land surveying (measuring curved boundaries).

Degrees vs. Radians

Degrees divide a full rotation into 360 equal parts — a convention dating back to Babylonian astronomy. Radians, on the other hand, define angles based on the ratio of arc length to radius, making them the natural unit for calculus and physics. One radian equals approximately 57.296 degrees. The conversion formulas are: degrees = radians × 180/π and radians = degrees × π/180. This calculator accepts either unit and displays both, so you never need to convert manually. In scientific contexts, radians are almost always preferred because they simplify derivative formulas: for example, d/dx sin(x) = cos(x) only works when x is in radians.

Chord Length

The chord is the straight-line distance between the two endpoints of an arc. Using the law of cosines or basic trigonometry, the chord length is c = 2r sin(θ/2). For small angles, the chord length is approximately equal to the arc length (both approach rθ as θ approaches zero). The difference between arc length and chord length matters in road design (horizontal curves), satellite orbit calculations, and geodesy, where the curvature of the Earth makes straight-line distances shorter than along-surface distances. This calculator shows both values so you can compare them directly.

Real-World Applications

Arc length calculations appear in many practical contexts. In road and railway engineering, horizontal curves are designed with specific radii and central angles, and the arc length determines the length of the curved road section. In manufacturing, CNC machines cut along arcs and need precise arc lengths for toolpath programming. In physics, the arc length traversed by a pendulum or a planet in orbit is computed using these formulas. In navigation, great-circle distances between points on Earth are arc lengths on a sphere. Clock hands sweep sectors whose areas and arc lengths can be calculated with these formulas. Even in cooking, when cutting a pie into equal slices, the arc length of each slice edge is determined by dividing the full circumference by the number of slices.