Cycloid Calculator

What Is a Cycloid?

A cycloid is the curve traced by a fixed point on the rim of a circle as it rolls, without slipping, along a straight line. Belonging to the broader family of roulettes, the cycloid is the specific case where a circle rolls on a flat surface. First rigorously studied in the 17th century, it earned the nickname the Helen of Geometry for the fierce disputes it inspired among Europe's greatest mathematicians, including Pascal, Huygens, Newton, and the Bernoullis.

Parametric Equations of a Cycloid

The position of the tracing point is defined by two parametric equations, both depending on the rolling angle t (in radians) and the circle's radius r:

• Horizontal position: x = r(t − sin t)
• Vertical position: y = r(1 − cos t)

As t advances from 0 to 2π, the circle completes one full revolution and the tracing point sweeps a single arch, departing from and returning to the base line. Each subsequent 2π interval produces one more identical arch. At t = 0, both x and y equal zero. At t = π, the point reaches its maximum height at the apex of the arch. At t = 2π, it touches down again at horizontal distance 2πr from the start.

Arc Length of One Arch: L = 8r

The total length of one complete cycloid arch equals exactly eight times the radius of the generating circle:

L = 8r

For a circle with radius r = 5 cm, one arch is 40 cm long — four times the circle's diameter. Christopher Wren derived this result in 1658. Remarkably, the formula contains no π, making it one of the cleanest results involving circular motion.

The derivation applies the parametric arc-length integral: L = integral from 0 to 2π of √((dx/dt)² + (dy/dt)²) dt. Computing derivatives gives dx/dt = r(1 − cos t) and dy/dt = r sin t. The integrand simplifies to r√(2 − 2cos t) = 2r|sin(t/2)| via the half-angle identity, and integrating over [0, 2π] yields 8r. A complete derivation appears in Section 12.5 of Whitman College's online Calculus text.

Area Under One Arch: A = 3πr²

The area enclosed between a single cycloid arch and the base line equals three times the area of the generating circle:

A = 3πr²

For r = 5 cm, the enclosed area is 3π × 25 ≈ 235.6 cm². Gilles de Roberval first established this result around 1634 using Cavalieri's indivisibles. The integral-calculus derivation applies the parametric area formula: A = integral from 0 to 2π of y · (dx/dt) dt = r² · integral from 0 to 2π of (1 − cos t)² dt. Expanding the square gives 1 − 2cos t + cos²t; integrating each term over a full period yields 3π, so A = 3πr². This technique is covered in detail at Paul's Online Math Notes, Calculus II: Area with Parametric Equations.

Variable Reference

• r — Radius of the rolling circle. Any consistent length unit (mm, cm, m, in) is valid. The value must be a positive real number. Doubling r doubles the arch length and quadruples the arch area.
• t — Parameter angle in radians. One complete arch spans t ∈ [0, 2π]. At t = π/2 the point has risen to height r(1 − cos(π/2)) = r; at t = π it reaches maximum height 2r; at t = 3π/2 it descends back to height r.
• x — Horizontal distance of the tracing point measured from its starting position on the base line.
• y — Height of the tracing point above the base line. The maximum value is 2r and occurs at t = π.

Real-World Applications

The Brachistochrone: Fastest Descent Curve

In 1696, Johann Bernoulli challenged mathematicians to find the curve along which a bead slides fastest between two points under gravity. Five mathematicians — Newton, Leibniz, L'Hôpital, Jakob Bernoulli, and Johann Bernoulli himself — independently proved the answer is an inverted cycloid arch. A ball on a cycloidal ramp reaches the bottom faster than on any straight incline, circular arc, or other path.

The Tautochrone: Equal Descent Times

An inverted cycloid is also a tautochrone: a ball released from any point on the arch reaches the lowest point in the same elapsed time, regardless of starting height. Christiaan Huygens exploited this property in 1673 to design cycloidal cheeks for pendulum clock escapements, correcting amplitude-dependent period variation. Both properties are discussed in the University of Texas at Austin's M408D treatment of the cycloid.

Gears, Rotors, and Engineering Design

Cycloidal tooth profiles appear in precision watch movements and planetary gear trains, where they minimize friction and backlash. The Wankel rotary engine uses an epitrochoid — a close relative of the cycloid — as its rotor housing curve. Structural engineers also employ cycloidal arch shapes in bridge ribs and tunnel cross-sections for favorable load distribution.

Authoritative Sources

• The Cycloid — University of Texas at Austin, M408D
• Calculus with Parametric Equations, Sec. 12.5 — Whitman College
• Area with Parametric Equations — Paul's Online Math Notes
• Cycloids and Paths — Portland State University