Arc Length Calculator

Arc length is the distance along a curved line that forms part of a circle's circumference. To calculate arc length, use the formula s = rθ, where s is the arc length, r is the radius, and θ is the central angle in radians. If your angle is in degrees, first convert it to radians by multiplying by π/180, or use the alternative formula s = (θ/360) × 2πr. For example, a circle with radius 10 and a 90° central angle has arc length = 10 × (90 × π/180) = 10 × 1.5708 ≈ 15.71 units.

To convert degrees to radians, multiply the angle in degrees by π/180. For example, 90° = 90 × (π/180) = π/2 ≈ 1.5708 radians. To convert radians to degrees, multiply by 180/π. For example, π radians = π × (180/π) = 180°. Common conversions to memorize: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 270° = 3π/2, 360° = 2π. A full circle is 360° or 2π radians, and this relationship is the basis for all conversions.

Arc length is the curved distance along the circumference between two points on a circle, while chord length is the straight-line distance between those same two points. The arc is always longer than or equal to the chord (they're equal only when both are zero). Arc length formula: s = rθ. Chord length formula: c = 2r × sin(θ/2). For small angles, arc and chord are nearly equal. As the angle increases toward 180°, the difference grows. At 180°, the chord equals the diameter (2r), while the arc equals half the circumference (πr).

Sector area is the region enclosed by two radii and an arc of a circle (like a pizza slice). The formula is A = (1/2) × r² × θ, where r is the radius and θ is the central angle in radians. Alternatively, using degrees: A = (θ/360) × πr². For example, a sector with radius 10 and central angle 90° has area = (1/2) × 10² × (π/2) = 50 × 1.5708/2 ≈ 78.54 square units. This represents 1/4 of the full circle's area (since 90° is 1/4 of 360°).

Sector perimeter (also called sector perimeter or arc perimeter) is the total length of the boundary of a sector, which includes two radii plus the arc length. The formula is P = 2r + s = 2r + rθ = r(2 + θ), where r is the radius, s is the arc length, and θ is the central angle in radians. For a sector with radius 10 and 90° (π/2 radians) central angle: P = 10 × (2 + π/2) = 10 × 3.5708 ≈ 35.71 units. This is useful when calculating fencing for pie-shaped plots or material for fan-shaped designs.

Arc length calculations are essential in many fields: (1) Engineering: designing curved roads, railway tracks, gear teeth, and cam profiles. (2) Architecture: planning curved walls, domes, and arches. (3) Physics: calculating distances in circular motion, pendulum swings, and planetary orbits. (4) Navigation: determining distances along Earth's surface using great circle routes. (5) Manufacturing: cutting curved materials, designing rounded corners, and creating circular patterns. (6) Sports: measuring running track lanes (each outer lane is longer due to increased arc length). (7) Animation: calculating smooth curved motion paths.

Arc length connects linear distance to angular measurement. The relationship is s = rθ, which differentiates to v = rω, where v is linear velocity (speed along the arc), r is radius, and ω is angular velocity (rate of angle change in radians per second). This means an object at a larger radius travels a greater arc length in the same time, even if the angular velocity is constant. For example, on a merry-go-round, horses at the outer edge move faster (higher linear speed) than those near the center, despite rotating at the same angular velocity. This principle is fundamental in rotational mechanics and is used in designing wheels, gears, and turbines.