Free Cycloid Calculator

Introduction to the Cycloid Parameters Calculator

The Cycloid Parameters Calculator (a free cycloid curve calculator available online) gives you instant access to all essential measurements of a cycloid curve. Simply enter the radius $r$ of the generating circle, and the tool computes the arc length, hump length (the circumference of the rolling circle), hump height (its diameter), the area under a single arch, and the full perimeter of that arch. Whether you need the cycloid arc length area for a physics problem or the curve’s dimensions for a design project, this web‑based instrument streamlines the work.

Understanding the Cycloid

A cycloid is the path traced by a fixed point on the circumference of a circle as it rolls without slipping along a straight line. For example, imagine a dot on the rim of a wheel rolling along a road—the dot’s trajectory is a cycloid. The curve was named and methodically explored by Galileo after earlier mathematicians had claimed its discovery.

Mathematical Description and Key Formulas

The cycloid is most easily described with parametric equations. If the rolling circle has radius $r$ and rotates through an angle $\theta$ (in radians), the coordinates of the tracing point are:

$$x = r (\theta - \sin \theta), \qquad y = r (1 - \cos \theta)$$

The corresponding Cartesian equation (obtained by eliminating $\theta$) is:

$$x = r \cos^{-1}\!\left(1 - \frac{y}{r}\right) - \sqrt{y\,(2r - y)}$$

These equations yield all the geometric properties of one complete arch (a full rotation of the circle):

• Arc length ($S$): the length of the curved portion between two consecutive cusps.

  $$S = 8r$$

  The arc length depends solely on the radius and is surprisingly independent of $\pi$.

• Hump length ($C$): the straight‑line distance between consecutive cusps. This is exactly the circumference of the rolling circle:

  $$C = 2\pi r$$

• Hump height ($d$): the maximum vertical rise of the arch, equal to the rolling circle’s diameter:

  $$d = 2r$$

• Area under one arch ($A$): the region enclosed by the arch and the baseline. It works out to three times the area of the generating circle:

  $$A = 3\pi r^{2}$$

• Perimeter of one arch ($p$): the sum of the arc length and the basal hump length:

  $$p = C + S = 2\pi r + 8r$$

When you provide the radius to the cycloid calculator, all these parameters are returned instantly.

Step‑by‑Step Construction of a Cycloid

To draw a cycloid manually, use the following procedure (illustrated here for a circle of radius 25 mm):

1. Draw the generating circle with the chosen radius (e.g., 25 mm) and mark its centre as $C$.
2. Create the baseline: a horizontal line of length $2\pi r$ (about 157 mm for $r=25$ mm). Also draw a parallel horizontal line of the same length through centre $C$.
3. Divide both horizontal lines into 12 equal segments. Label the points on the centre line $C_1, C_2, \dots, C_{12}$ and the points on the baseline $1, 2, \dots, 12$.
4. Join corresponding points: draw straight lines from $C_i$ to $i$.
5. From each division point on the circle, draw a horizontal line towards the right (parallel to the centre line).
6. Using each $C_i$ as a centre and the original radius $r$, draw an arc that intersects the horizontal line passing through point $i$. This creates a set of intersection points.
7. Connect the intersection points with a smooth freehand curve. The resulting arch is exactly one cycloid arch.

Repeating the steps generates subsequent arches, each identical to the first.

Properties and Practical Use

The cycloid is a periodic curve: it repeats its shape with every full roll of the circle. The vertical component of a point moving along a cycloid follows simple harmonic motion, which makes the curve relevant in mechanical vibrations and pendulum designs. A notable large‑scale application is the Kimbell Art Museum in Texas, where the ceiling employs cycloidal vaults.

Variations of the Cycloid

The standard cycloid assumes the tracing point lies on the circle’s rim. Changing the point’s location gives rise to other curves:

• Curtate cycloid: the point is inside the circle (e.g., on a spoke). The arch does not touch the baseline and has no cusps.
• Prolate cycloid: the point is outside the circle. The arch includes loops that go below the baseline.
• Trochoid: a generic term for curves generated by a point at a fixed distance from the centre of a rolling circle (encompassing curtate and prolate).
• Hypocycloid: the rolling circle turns inside a larger fixed circle, producing star‑shaped paths.
• Epicycloid: the rolling circle rotates externally around another circle, generating petal‑like curves.
• Hypotrochoid and epitrochoid: trochoid variants for internal and external rolling, respectively.

While these variants require different parameterisations, this tool focuses on the standard cycloid where the point is on the circumference.