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.
Use this when a circle problem mixes curved distance and straight-line distance and you need to see how arc length, chord length, sagitta, and sector area change together. It is practical for geometry coursework, machining layouts, and curved-structure design because the angle and radius stay attached to every derived length.
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.
Calculate arc length, chord length, and circumference of a circle. Supports degrees and radians, with sagitta, sector area, and arc-to-chord ratio. Use it when you need a repeatable calculation in the math / geometry category and want the setup, result, and supporting values kept together. This is especially helpful when small input changes, unit choices, or rounding decisions can change the final number.
Start by confirming that the inputs match the formula shown on the page. Then compare the main output with the worked example and any secondary values shown by the calculator. If the result will be used in another calculation, keep extra precision until the final step and record the assumptions beside the number.
Treat the result as a calculation aid rather than a substitute for context. For schoolwork, include the formula and substitution steps. For planning, technical, financial, or health-related decisions, verify important numbers against primary records, current rules, or a qualified professional before acting on them.
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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.
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