Cycloid Calculator

Enter the rolling circle's radius and roll angle to get the traced point's position (x, y), the arc length, and the area under one arch using x = r(θ − sin θ) and y = r(1 − cos θ).

Quick Facts

Parametric equations
x = r(θ − sin θ), y = r(1 − cos θ)
θ is the angle (in radians) the circle has rolled through.
Arc length (one arch)
L = 8r
First proven by Christopher Wren in 1658.
Area under one arch
A = 3πr²
Exactly three times the area of the generating circle.
Max height
h = 2r
Reached at θ = π, equal to the circle's diameter.

Your Results

Calculated
Point Position (x, y)
-
Traced point's location at angle θ
Arc Length
-
L = 8r per arch
Area Under Curve
-
A = 3πr² per arch
Max Height
-
h = 2r, constant per arch

Ready

Enter a radius and roll angle, then press Calculate.

Formula and Method for the Cycloid Calculator

A cycloid is the curve traced by a single fixed point on the rim of a circle of radius r as that circle rolls, without slipping, along a straight line. As the circle turns through an angle θ (measured in radians), its center advances horizontally by rθ while the rim point sweeps around the center. Combining that translation with the point's rotation gives the parametric equations x(θ) = r(θ − sin θ) and y(θ) = r(1 − cos θ). This calculator uses those equations to find the traced point's position, plus the classic arc-length and area results for a full arch.

How the calculation works

Enter the radius of the rolling circle and its unit, then a roll angle θ between 0° and 360°. The calculator converts θ to radians and plugs it into x(θ) = r(θ − sin θ) and y(θ) = r(1 − cos θ) to locate the traced point. One complete arch corresponds to θ running from 0 to 2π (360°): its arc length is L = 8r and the area enclosed between the arch and the line is A = 3πr² — both derived by integrating the parametric equations over a full revolution. The arch's height is constant at h = 2r, reached at θ = 180° (the top of the arch). If you set "Number of Arches" above 1, the arc length and area scale linearly, matching a chain of identical arches laid end to end (the pattern traced by a wheel rolling repeatedly).

Common mistakes

  • Angle units: the calculator takes θ in degrees and converts internally — do not enter radians directly, or the point position will be wrong.
  • Confusing radius with diameter: r in the formulas is the radius of the rolling circle, not its diameter; the arch height (2r) equals the diameter, which can cause mix-ups.
  • Arc length vs. base width: the arc length of one arch (8r) is longer than the arch's horizontal base (2πr ≈ 6.28r) because the curve bows upward between the two.

Real-world applications

  • Cycloidal gear teeth and pin-wheel (cycloidal) gear profiles use the same rolling-circle geometry to minimize friction and wear.
  • The cycloid is the solution to the brachistochrone problem — the shape of a frictionless wire that lets a bead slide between two points in the least time.
  • It also solves the tautochrone problem: a ball released from any point on an inverted cycloid reaches the bottom in the same amount of time.
  • Cycloidal drives are used in robotics and precision gearboxes for high-ratio, low-backlash speed reduction.

Frequently Asked Questions

What is a cycloid?
A cycloid is the curve traced by a fixed point on the edge of a circle of radius r as the circle rolls without slipping along a straight line. It is described by the parametric equations x = r(θ − sin θ) and y = r(1 − cos θ), where θ is the angle the circle has rotated through.
What is the arc length of one arch of a cycloid?
One complete arch, traced as θ runs from 0 to 2π, has an arc length of L = 8r, where r is the radius of the rolling circle. Christopher Wren proved this result in 1658.
What is the area under one arch of a cycloid?
The area between one arch of the cycloid and the line it rolls on is A = 3πr², exactly three times the area of the generating circle (πr²).
How tall is a cycloid arch?
The maximum height of each arch is h = 2r, reached at θ = π (the halfway point of the roll), which equals the diameter of the rolling circle.

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