Cycloid Calculator

Calculate properties of a cycloid curve: arc length, area under one arch, and parametric coordinates using x = r(t - sin t), y = r(1 - cos t).

Enter Cycloid Parameters

Result

Area Under One Arch
235.619449
square units
Arc Length (one arch)40
Arc Length (all cycles)40
Total Area (all cycles)235.619449
Base Length (one arch)31.415927
Max Height10
x(t) at given t15.707963
y(t) at given t10

Step-by-Step Solution

Area = 3 pi r^2 = 3 pi (5)^2 = 235.619449

Understanding the Cycloid Curve

A cycloid is the curve traced by a point on the rim of a circular wheel as it rolls along a straight line without slipping. It is one of the most beautiful and historically significant curves in mathematics. The cycloid was studied extensively by Galileo, Pascal, Bernoulli, and many other famous mathematicians, earning it the nickname "the Helen of geometry" due to the many disputes it caused among scholars.

The cycloid is defined parametrically: for a circle of radius r rolling along the x-axis, a point on the rim traces the path given by x = r(t - sin t) and y = r(1 - cos t), where t is the angle of rotation in radians. One complete arch corresponds to t going from 0 to 2pi.

Cycloid Formulas

Parametric Equations

Position of the tracing point as the wheel rotates by angle t.

x = r(t - sin t), y = r(1 - cos t)

Area Under One Arch

The area between one complete arch and the baseline.

A = 3 pi r2

Arc Length (One Arch)

The total length of the curve for one complete cycle.

L = 8r

Base Length (One Arch)

The horizontal distance covered by one full rotation.

b = 2 pi r

Maximum Height

The highest point of the arch, reached at t = pi.

h_max = 2r

Curvature at Top

The radius of curvature at the highest point of the arch.

R_curv = 4r (at t = pi)

The Brachistochrone Problem

One of the most famous results involving the cycloid is the brachistochrone problem, posed by Johann Bernoulli in 1696. The problem asks: what is the curve along which a bead sliding under gravity (without friction) will travel from one point to another in the least time? The answer, remarkably, is an inverted cycloid. This result was independently discovered by Newton, Leibniz, L'Hopital, and both Bernoulli brothers.

The Tautochrone Property

The cycloid also has the remarkable tautochrone property: a bead released from any point on an inverted cycloid will reach the bottom in exactly the same time, regardless of its starting position. This property was discovered by Christiaan Huygens in 1659, who used it to design a pendulum clock with cycloid-shaped cheeks to ensure isochronous (equal-time) oscillations.

Area Ratio

The area under one arch of a cycloid is exactly 3 times the area of the generating circle. This elegant result was first proven by Gilles de Roberval in 1634. Since the generating circle has area pi r2, the arch area is 3 pi r2. This means a cycloid arch encloses exactly three times the area of the wheel that generates it.

Arc Length Derivation

The arc length of one cycloid arch is exactly 8r, which is four times the diameter of the generating circle. This can be derived using the arc length formula for parametric curves: L = integral from 0 to 2pi of sqrt((dx/dt)2 + (dy/dt)2) dt. After simplification using trigonometric identities, the integral evaluates to 8r.

Applications of Cycloids

  • Clock Design: Huygens used cycloid curves in pendulum clocks for improved accuracy.
  • Gear Teeth: Cycloidal curves are used in the design of gear tooth profiles.
  • Architecture: Cycloid arches appear in some architectural designs for aesthetic and structural reasons.
  • Physics: The brachistochrone and tautochrone properties are fundamental in classical mechanics.
  • Optics: Cycloid curves appear in the study of wave phenomena and caustics.