Calculate arc length, chord length, and circumference of a circle. Supports degrees and radians, with sagitta, sector area, and arc-to-chord ratio.
When working with circles, three related but distinct "length" measurements come up constantly: the circumference (the full perimeter), the arc length (a curved portion of the perimeter), and the chord length (the straight line connecting two points on the circle). This calculator handles all three and shows how they relate to each other.
The circumference is C = 2πr — the total distance around the circle. An arc length is the distance along a curved section defined by a central angle: L = (θ/360°) × 2πr in degrees, or simply L = rθ in radians. A chord is the straight line between the two endpoints of the arc: chord = 2r sin(θ/2). The arc is always longer than the chord for the same angle (except at 0°), and their ratio approaches 1 as the angle approaches zero — a fundamental fact in calculus.
The calculator also computes the sagitta (the height of the arc above the chord), the sector area, the arc-to-chord ratio, and the fraction of the full circle the arc represents. It supports both degree and radian input and includes a reference table of common arc angles with pre-computed factors. Engineers use arc and chord calculations for bridge design, gear teeth, cam profiles, and any application involving circular motion or curved structures.
This circle length calculator reduces manual rework when you need quick checks for assignments, exam prep, and design calculations. You can enter Primary Calculation, Unit, Radius, Angle Unit and immediately see dependent measurements, validity checks, and geometry relationships in one place. That makes it easier to catch input mistakes early and confirm your final answer before moving to the next step.
Circumference: C = 2πr Arc length (degrees): L = (θ/360) × 2πr Arc length (radians): L = rθ Chord length: chord = 2r sin(θ/2) Sagitta: h = r(1 − cos(θ/2)) Sector area: S = (θ/360) × πr² Arc/chord ratio = L / chord (→ 1 as θ → 0)
Result: For radius=10, angle=90, angleUnit=degrees, the tool returns the solved circle length outputs shown in the result cards.
This example uses a realistic input set from the calculator workflow. After entry, the calculator applies the built-in circle length formulas and reports derived values, checks, and classifications automatically.
This page is tailored to circle length, with outputs tied directly to the form fields (Primary Calculation, Unit, Radius, Angle Unit). Instead of a one-line formula dump, it consolidates validation, derived metrics, and interpretation so you can solve and verify in one pass.
Use this tool for homework checks, worksheet generation, tutoring walkthroughs, and quick engineering geometry estimates. Presets and visual output blocks make it easier to compare scenarios and understand how each input affects the final result.
Keep units consistent, match each value to the correct field, and watch validity indicators before using the final numbers. If your case looks off, change one input at a time and use the output details to identify the mismatch quickly.
Arc length is the distance along a curved portion of the circle's circumference, defined by a central angle. Formula: L = (θ/360) × 2πr in degrees.
A chord is a straight line connecting two points on the circle. The longest chord is the diameter. Chord length = 2r sin(θ/2) for central angle θ.
The sagitta (or arc height) is the perpendicular distance from the midpoint of a chord to the arc. h = r(1 − cos(θ/2)).
Yes, for any angle greater than 0° and less than 360°. The arc follows the curve while the chord takes the straight path. Their ratio approaches 1 for very small angles.
Multiply degrees by π/180 to get radians. Multiply radians by 180/π to get degrees. For example, 90° = π/2 ≈ 1.5708 radians.
When the central angle is 60°. This is because 2 sin(30°) = 1, so chord = 2r × 0.5 = r. The triangle formed is equilateral.