Chord Finder - Circle Chord Length & Arc Calculator

The Chord Finder calculator is a precision geometric tool designed to solve circle-related measurement problems instantly. Given a circle's radius and a central angle, it computes four interconnected measurements: chord length, arc length, sector area, and segment area. These quantities appear throughout engineering, architecture, manufacturing, and mathematics, making a reliable calculator an essential resource for professionals and students alike. A chord is a straight line segment whose two endpoints both lie on the circumference of a circle. The length of a chord depends entirely on two things: the circle's radius and the central angle subtended by that chord. The fundamental formula is c = 2r × sin(θ/2), where r is the radius and θ is the central angle measured in degrees. Because sine is a non-linear function, equal increments in angle do not produce equal increments in chord length — a relationship that surprises many first-time users. Arc length measures the curved distance along the circle's edge between the same two endpoints that define the chord. While the chord cuts straight across, the arc follows the circle's boundary. The formula is s = r × θ_rad, where θ_rad is the central angle converted to radians (θ × π/180). For a full circle (360°), this reduces to the familiar circumference formula C = 2πr. Sector area is the area of the pie-slice region bounded by two radii and the arc. It is computed as A_sector = (θ/360) × πr², representing the proportion of the full circle's area that the sector occupies. For a 90° angle, the sector is exactly one quarter of the circle. Segment area is the region between the chord and the arc — a crescent-shaped area. To find it, subtract the triangular area formed by the two radii and the chord from the sector area: A_segment = A_sector − (½)r²sin(θ). When θ = 180°, the segment becomes a semicircle because the triangle degenerates to zero area. Engineers use chord and arc calculations when designing arch bridges, curved road transitions, pipeline bends, and gear tooth profiles. Architects rely on them for circular windows, domed ceilings, and curved facades. Machinists apply the formulas to circular jig plates, cam profiles, and precision cutting operations. Students encounter these relationships in trigonometry, calculus, and analytic geometry courses. All inputs are validated to ensure r > 0 and 0 < θ ≤ 360°. Results are displayed to four decimal places, which provides sufficient precision for most engineering and mathematical applications. For angles very close to 0° or 360°, floating-point arithmetic remains accurate because sin() is numerically stable near these values.