What is a Cycloid?
Try to picture a dropped coin rolling along a straight line. Now imagine a point anywhere on the edge of that coin. That point traces a curve as it moves — that curve is a cycloid. In more precise terms, a cycloid is a curve formed by a circle rotating along a straight line without slipping.
Galileo was the first to study the curve methodically and gave it the name cycloid. The history of the cycloid is a rich one: many famous mathematicians — including Pascal, Descartes, Roberval, Huygens, and the Bernoullis — studied and argued over its properties.
Cycloid Equation
A cycloid is fully defined by the radius (r) of the rolling circle. The parametric (polar) equations are:
x = r × (θ − sin θ)y = r × (1 − cos θ)
where r is the radius and θ is the angular displacement of the rolling circle.
The Cartesian equation is:
x = r × cos⁻¹(1 − y/r) − √(y × (2r − y))
Cycloid Properties & Formulas
| Property | Formula | Description |
|---|---|---|
| Arc Length (S) | S = 8 × r |
Curved distance between two cusps (one full arch) |
| Area (A) | A = 3 × π × r² |
Area enclosed under one complete arch |
| Hump Length (C) | C = 2 × π × r |
Straight-line distance between two cusps — equals the circumference of the rolling circle |
| Hump Height (d) | d = 2 × r |
Height of the arch — equals the diameter of the rolling circle |
| Perimeter (p) | p = C + S |
Sum of the arc length and the hump length |
How to Construct a Cycloid
To construct a cycloid on paper using geometric tools:
- Draw a circle with the chosen radius (e.g., 2.5 cm) and mark its centre C.
- Draw a horizontal line from the base of the circle equal in length to the circle's circumference (2πr).
- Divide both the circle and the horizontal line into 12 equal parts. Label the points on the horizontal line C₁, C₂ … C₁₂.
- Connect each centre position Cₙ to the corresponding point n on the baseline.
- Draw horizontal lines from each division point on the circle in the direction of rolling.
- From each centre Cₙ, draw an arc of radius r crossing the corresponding horizontal line. Mark the intersection point.
- Join all marked intersection points with a smooth freehand curve — that is your cycloid.
Cycloid Curve
A cycloid curve follows simple harmonic motion — it is periodic and repeats an identical pattern with each full revolution. The cycloid is notable for two famous properties:
- Tautochrone: A ball placed anywhere on a cycloidal surface reaches the bottom in the same time, regardless of starting position.
- Brachistochrone: The cycloid is the curve of fastest descent under gravity between two points not on the same vertical line.
The Kimbell Art Museum in Fort Worth, Texas uses cycloidal arches in its iconic roof design.
Unit Systems
This calculator supports both metric and imperial measurement systems:
- Metric: mm (millimeters), cm (centimeters), m (meters), km (kilometers)
- Imperial: in (inches), ft (feet), yd (yards), mi (miles)
All output values — arc length, hump length, hump height, and perimeter — are expressed in the same unit as the input radius. The area is expressed in the square of that unit (e.g., cm²).
Variations of a Cycloid
A standard cycloid traces a point on the rim of the rolling circle. Moving the tracing point gives different curves:
- Curtate cycloid — tracing point is inside the circle (flattened arches)
- Prolate cycloid — tracing point is outside the circle (loops appear)
- Trochoid — general name for curtate and prolate cycloids
- Hypocycloid — circle rolls inside another circle
- Epicycloid — circle rolls outside another circle
- Hypotrochoid — trochoid inside another circle
- Epitrochoid — trochoid outside another circle
FAQs
How do I find the arc length of a cycloid?
Use the formula S = 8 × r, where r is the radius of the rolling circle.
How do I find the area under a cycloid arch?
Use the formula A = 3 × π × r². Interestingly, this is exactly 3 times the area of the rolling circle itself (πr²).
What is the hump length of a cycloid?
The hump length is the straight-line distance between two consecutive cusps. It equals the circumference of the rolling circle:
C = 2 × π × r.
What is the hump height?
The hump height is the maximum height of the cycloid arch, equal to the diameter of the rolling circle: d = 2r.
What is the perimeter of a cycloid arch?
The perimeter is the total boundary length of one arch: arc + base.
p = C + S = 2πr + 8r = r(2π + 8).