Cycloid Calculator - One Arch from the Rolling Circle

A cycloid calculator is a geometry tool that turns the radius of a rolling circle into every other dimension of one cycloid arch, including the area under the arch, the curved arc length, the straight hump length, the hump height, and the perimeter of one full arch. The cycloid is the curve traced by a fixed point on the rim of a circle that rolls without slipping along a straight line, so a single positive radius fully determines the shape.

• • Geometry students: Check homework and exam problems for the area of a cycloid or the arc length of one arch without redoing the derivations.
• • Physics students: Set up the brachistochrone and tautochrone problems, where the cycloid is the curve of fastest descent and equal-time roll.
• • Architects and designers: Explore the cycloidal arch used in historical and modern designs, from the Kimbell Art Museum to vault profiles.
• • Engineers: Estimate the curved length of a cycloidal cam profile or the swept area of a cycloidal gear in a quick design sketch.

Once you type the radius of the generating circle, the cycloid calculator returns the area, arc length, hump length, hump height, and perimeter of one full arch in the same unit. You can switch between millimeters, centimeters, meters, inches, and feet so the same number works for a classroom diagram or a fabrication drawing.

The cycloid shares its rolling-curve pedigree with the hanging chain, so the catenary curve calculator is a natural companion for any graph of y = a cosh(x/a).

The calculator only needs the rolling circle's radius. From that single value it applies five classical cycloid formulas to give you the area under one arch, the arc length, the hump length, the hump height, and the perimeter of one full arch.

• r: Radius of the rolling circle, entered in the chosen length unit.
• A: Area under one arch of the cycloid, equal to 3πr².
• S: Curved arc length of one arch, equal to 8r.
• C: Hump length, the straight-line distance between two cusps, equal to 2πr.
• d: Hump height, equal to the diameter 2r of the rolling circle.
• p: Perimeter of one arch, the sum of the hump length and the arc length.

Every formula is evaluated from the same radius in the same unit, so the area is reported in the squared unit (cm², m², in², ft²) and the rest stay in the linear unit. Use the result as a single-number check for textbook answers or as the input for downstream calculations such as cam profile length, swept area, or arch volume.

According to Wolfram MathWorld, the area under one arch equals 3πr² and the arc length of one arch equals 8r

For a circular arc of the same span, the arc length calculator gives the related s = rθ value, which is a useful comparison against the cycloid's S = 8r.

These four cards cover the vocabulary that shows up the moment a cycloid is plotted, traced on paper, or used in a physics derivation.

If you only remember one formula, remember S = 8r: the arc length of a cycloid arch is always exactly eight times the generating radius, no matter the units you work in. The area formula A = 3πr² has the same pleasing exactness, which is why textbook problems rarely need approximation. The same curve can be written in parametric form as x = r(θ - sin θ) and y = r(1 - cos θ), with θ the angle (in radians) turned by the rolling circle.

Because the cycloid is generated by a circle, the circle calculator is the right place to confirm the circumference and area of the underlying wheel in the same units.

Five quick steps turn a single radius into every cycloid dimension. The form works on phones, tablets, and desktop browsers without any setup.

1. 1 Enter the radius: Type the radius of the rolling circle in the first field. Use 0 or a positive number, with up to four decimal places for finer drawings.
2. 2 Pick the length unit: Select mm, cm, m, in, or ft so every linear result uses the same unit and the area is reported in its square.
3. 3 Read the area result: The first row of the results panel shows the area under one arch in the squared unit. Use it for swept-area and volume problems.
4. 4 Read the arc, hump, and height: The next three rows give the curved arc length, the straight hump length between cusps, and the hump height. Each is a direct formula of the radius.
5. 5 Use the perimeter for the outline: The perimeter row adds the arc length and hump length, which is the total outline of one cycloid arch. Use it for material cut lists or profile lengths.

Try radius = 7 cm with the cm unit: the calculator reports A = 461.8141 cm², S = 56 cm, C = 43.9823 cm, d = 14 cm, and p = 99.9823 cm. That is enough to lay out a cycloidal arch on graph paper in a few seconds.

When you want to compare a cycloid arch to a smoother oval outline, the ellipse area calculator returns the same kind of area in a closed form for the matching semi-axes.

The calculator saves time on derivations, removes unit mistakes, and keeps every cycloid dimension consistent for downstream work.

• • Five answers in one pass: You enter the radius once and immediately get the area, arc length, hump length, hump height, and perimeter of one arch.
• • Exact classical formulas: The numbers come from the S = 8r, A = 3πr², and C = 2πr results that have been known since the seventeenth century.
• • No unit conversion mistakes: Pick the unit once and every linear output uses it, while the area uses its square. The same number works in mm, cm, m, in, and ft.
• • Useful for cam and gear design: Cycloidal cam profiles and cycloidal gear tooth flanks rely on the same arc length, so the calculator gives a quick first-pass check.
• • Cross-validation for homework: The worked example for r = 5 cm matches the reference cycloid example, so students can confirm their hand calculations in seconds.

Because the formulas are closed form, the results do not depend on numerical integration, meshing, or iterative solvers. That makes the cycloid calculator reliable as a teaching reference and as a sanity check for design sketches that approximate the cycloid with splines or arcs.

If you are sizing a curve that lives next to a circular loop, the circle length calculator gives the corresponding s = 2πr outline in the same radius context.

All five outputs scale directly with the radius you enter, so the only thing that changes the numbers is the radius value, the unit you select, and the precision of your input.

• • The result covers exactly one arch of an ideal cycloid. Real physical curves, like a coin rolling on a table, deviate slightly because of slip, finite contact patch, and surface roughness.
• • The calculator assumes the rolling circle moves in a vertical plane without tilting. Curves traced on a 3D path or a non-flat base follow different equations and need a different tool.

Treat the result as the textbook value for one arch. For cam profiles, gears, and architectural arches, add a small safety margin to cover tolerance, friction, and measurement rounding.

According to Wikipedia, hump length C = 2πr and hump height d = 2r for one arch

If you want a half-circle baseline to compare with the cycloid's hump length, the semicircle area calculator gives the matching area and perimeter in a simpler closed form.