Calculate cycloid curve properties: arc length, area under arch, cusps, parametric points, surface of revolution. Includes interactive curve plot and properties table.
A cycloid is the curve traced by a point on the rim of a circle as it rolls along a straight line without slipping. Despite its simple genesis, the cycloid has remarkable mathematical properties that captivated mathematicians for centuries — Galileo, Pascal, Bernoulli, and many others studied it. The arc length of one arch is exactly 8r (eight times the radius), and the area under one arch is 3πr² (three times the area of the generating circle). These elegant closed-form results make the cycloid a favorite in calculus courses. Beyond pure mathematics, the cycloid is the solution to two famous physics problems: the brachistochrone (the curve of fastest descent under gravity) and the tautochrone (the curve where a sliding object reaches the bottom in the same time regardless of starting point). This calculator computes all key properties for any radius and number of arches, evaluates parametric coordinates at any parameter value, and renders an interactive SVG curve showing cusps, peak height, and the rolling behavior. It also computes the surface area obtained by revolving one arch about the baseline.
The cycloid has surprisingly elegant closed-form properties (arc length = 8r, area = 3πr²) but computing parametric points, speed, and curvature at arbitrary parameters requires careful calculus. This calculator handles all of it instantly, renders the actual curve shape, and computes the surface area of revolution — a notoriously difficult integral. It is ideal for calculus and physics students exploring parametric curves, the brachistochrone problem, and curve properties.
x(t) = r(t − sin t) y(t) = r(1 − cos t) Arc length (1 arch) = 8r Area (1 arch) = 3πr²
Result: Arc length = 16, Area ≈ 37.70
For r = 2: Arc length = 16, area = 37.70, width = 12.57, max height = 4, 2 cusps per arch.
The brachistochrone problem asks: what curve between two points yields the fastest descent for a frictionless bead under gravity? Johann Bernoulli proved in 1696 that the answer is an inverted cycloid — not a straight line, and not a circular arc. The closely related tautochrone property states that a bead sliding down an inverted cycloid reaches the bottom in the same time regardless of where it starts. Huygens exploited this for isochronous pendulum clocks: cycloidal "cheeks" constrain the pendulum bob to follow a cycloid, making its period independent of amplitude.
The cycloid is defined parametrically as x(t) = r(t − sin t), y(t) = r(1 − cos t). The speed at parameter t is v(t) = 2r|sin(t/2)|, which is zero at the cusps (t = 0, 2π, ...) and maximum at the peak (t = π). Integrating this speed from 0 to 2π gives the arc length of one arch: ∫ 2r sin(t/2) dt = 8r. The area under one arch, computed by ∫ y dx = 3πr², is exactly three times the area of the generating circle — a result that amazed Galileo.
Cycloidal gear profiles (used in clocks and watches) have the advantage of constant-velocity ratio and smooth engagement. The shape of the teeth is an arc of an epicycloid or hypocycloid. In architecture, cycloid arches have been proposed as structurally efficient shapes. In particle physics, the cycloid describes the path of a charged particle in crossed electric and magnetic fields (the cycloid drift).
A cycloid is the curve traced by a fixed point on the edge of a circle rolling along a straight line. It consists of a series of arches separated by cusps.
A cusp is a point where the cycloid touches the baseline. The curve has a sharp corner there and the speed is zero.
This is derived by integrating the speed √((dx/dt)² + (dy/dt)²) = 2r|sin(t/2)| from 0 to 2π, which evaluates to 8r. Use this as a practical reminder before finalizing the result.
Finding the curve between two points that minimizes the travel time of a frictionless sliding bead under gravity. The answer is an inverted cycloid.
A curtate cycloid is traced by a point inside the rolling circle (shorter loops). A prolate cycloid is traced by a point outside (self-intersecting loops).
Cycloid profiles appear in gear tooth design (cycloidal gears), pendulum clocks (Huygens), and roller coaster loop design. Keep this note short and outcome-focused for reuse.