Chord Finder - Circle Chord Length & Arc Calculator
Calculate chord length, arc length, sector area, and segment area for any circle using radius and central angle.
Enter a radius and central angle (0–360°) to instantly compute chord length, arc length, sector area, and segment area using precise trigonometric formulas.
Chord Finder - Circle Chord Length & Arc Calculator
Calculate chord length, arc length, sector area, and segment area for any circle using radius and central angle.
About the Chord Finder 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.
Chord Length Examples
Representative calculations showing how chord length, arc length, and areas vary with radius and central angle.
| Radius / Angle | Chord / Arc | Notes |
|---|---|---|
| r = 10, θ = 60° | Chord = 10.0000 | Arc = 10.4720 | Equilateral triangle chord — chord equals radius. |
| r = 5, θ = 90° | Chord = 7.0711 | Arc = 7.8540 | Quarter-circle; chord = r√2 ≈ 7.0711. |
| r = 8, θ = 180° | Chord = 16.0000 | Arc = 25.1327 | Semicircle; chord equals the diameter. |
| r = 12, θ = 30° | Chord = 6.2117 | Arc = 6.2832 | Small angle: chord ≈ arc (both ≈ 6.28). |
How to Use the Chord Finder
- Enter the circle's radius in the Radius field — any positive number is accepted.
- Enter the central angle in degrees (between 0° and 360°) in the Central Angle field.
- Click Calculate to instantly see chord length, arc length, sector area, and segment area.
- Review the formula line below the results to understand how each value was derived.
- Click Reset to clear the inputs and calculate for a different circle or angle.
Chord Finder FAQ
What is the difference between a chord and an arc?
A chord is a straight line connecting two points on a circle, while an arc is the curved portion of the circle's circumference between those same two points. For the same central angle, the arc length is always greater than the chord length, except in the degenerate limit as the angle approaches zero, where the two converge.
Why does the chord equal the radius when the central angle is 60°?
When the central angle is 60°, the chord formula gives c = 2r × sin(30°) = 2r × 0.5 = r. The two radii and the chord form a perfect equilateral triangle, so all three sides are equal. This is a classic result in Euclidean geometry and a useful check for calculator accuracy.
What is the maximum possible chord length?
The maximum chord length is the diameter, which occurs when the central angle equals exactly 180°. At that point, the chord passes through the center of the circle and the formula gives c = 2r × sin(90°) = 2r × 1 = 2r, confirming that the chord equals the diameter.
How is segment area different from sector area?
Sector area is the pie-slice region bounded by two radii and the arc. Segment area is the smaller region bounded by the chord and the arc — it is the sector minus the triangle formed by the two radii and the chord. For a 180° angle, the triangle collapses to zero area and the segment equals the full semicircle.
Can I use this calculator for engineering tolerances?
Yes. The calculator outputs four decimal places, which matches standard engineering drafting precision. For tighter tolerances, verify with a full-precision computation tool. The formulas used are exact trigonometric relationships, so the only error source is the floating-point representation of π and sine values.
What happens at a central angle of 360°?
At exactly 360°, the chord length is zero (the two endpoints coincide at the same point), the arc length equals the full circumference 2πr, and both the sector area and the segment area equal the full circle area πr². The triangle degenerates to zero area, so the segment (the region between the chord and the arc) covers the entire circle. The calculator accepts 360° as a valid input and returns these mathematically correct values.