Cycloid Calculator — Arc Length, Area & Curve
Enter the radius of a rolling circle to find the arc length (8r) and area (3πr²) of each cycloid arch, and see the animated parametric curve.
enote
Length of one cycloid arch = 8r
- 1
Radius of rolling circle
1 - 2
Area per arch (3πr²)
3 × π × 1² = 9,4248 - 3
Arc length per arch (8r)
8 × 1 = 8One cycloid arch is exactly 8 times the rolling radius — proved by Christopher Wren in 1658.
Kako deluje ta kalkulator?
A cycloid is traced by a point on a rolling circle of radius r: x(θ) = r(θ − sin θ), y(θ) = r(1 − cos θ). One arch spans 2πr wide and 2r tall. Arc length of one arch = 8r (exact); area under one arch = 3πr² (exact, three times the circle area).
Formula
How this is calculated
A cycloid is the curve traced by a point on the rim of a circle of radius r as the circle rolls without slipping along a flat horizontal surface. Its shape is described by the parametric equations x(θ) = r(θ − sin θ) and y(θ) = r(1 − cos θ), where θ is the angle (in radians) through which the circle has rotated. One complete arch (one loop of the curve) is traced as θ goes from 0 to 2π, spanning a horizontal distance of 2πr and reaching a maximum height of 2r (the diameter of the rolling circle).
Two famous results make the cycloid historically significant. First, its arc length for one arch is exactly 8r — four times the diameter of the generating circle — derived by integrating the arc-length formula from the parametric equations. Second, the area under one arch is exactly 3πr², which is three times the area of the generating circle (πr²). Both results were proved in the 17th century by Roberval and Wren.
The cycloid is also the solution to two classic problems in physics: the brachistochrone (the path of fastest descent under gravity between two points) and the tautochrone (the curve where a ball reaches the bottom in the same time regardless of starting position). This calculator assumes an ideal, frictionless geometric construction and computes exact closed-form results — no numerical approximation is used for the arc length or area.
Pogosta vprašanja
The arc length of one arch is exactly 8r, where r is the radius of the rolling circle. This was first proved by Christopher Wren in 1658 and is four times the diameter of the generating circle.
The area enclosed between one arch of a cycloid and the flat baseline is exactly 3πr², which is three times the area of the rolling circle (πr²). This result was proved by Gilles de Roberval around 1634.
The cycloid is the brachistochrone — the fastest path for a ball to slide between two points under gravity — and the tautochrone — where balls released from any height reach the bottom in the same time. These properties are used in clock pendulum design and theoretical mechanics.
Znano tudi kot
TG we-Calculate Editorial Team. (2026). Cycloid Calculator — Arc Length, Area & Curve [Online calculator]. TG we-Calculate. https://we-calculate.com/sl/calculator/cycloid-calculator
TG we-Calculate Editorial Team. "Cycloid Calculator — Arc Length, Area & Curve." TG we-Calculate. 2026. https://we-calculate.com/sl/calculator/cycloid-calculator.
TG we-Calculate Editorial Team, "Cycloid Calculator — Arc Length, Area & Curve," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/sl/calculator/cycloid-calculator
@misc{wecalculate_cycloid_calculator, title = {Cycloid Calculator — Arc Length, Area & Curve}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/sl/calculator/cycloid-calculator}}, year = {2026}, note = {TG we-Calculate} }
Vam je ta kalkulator pomagal?