Cycloid Calculator
A cycloid is the elegant curve traced by a point on the rim of a circle rolling along a flat surface. Enter the circle radius and the rolling angle (in degrees or radians) to get the parametric coordinates, partial arc length, enclosed area, radius of curvature, and the full-arch properties: arc length, base width, height, and perimeter. Results update instantly as you type.
Formula
Worked example
For a circle of radius 5 cm: arc length of one arch S = 8 x 5 = 40 cm; area A = 3 x pi x 25 = 235.62 cm^2; hump length C = 2 x pi x 5 = 31.42 cm; height d = 10 cm. At t = 180 deg (pi rad), the tracing point is at x = 5(pi - 0) = 15.71 cm, y = 5(1-(-1)) = 10 cm (the apex), and the radius of curvature is 4 x 5 x |sin(90 deg)| = 20 cm.
What is a cycloid?
A cycloid is the curve traced by a fixed point on the rim of a circle as that circle rolls, without slipping, along a straight line. Imagine watching a chalk mark on a bicycle tire as the bike rolls forward: the mark rises from the ground in an arch, reaches the top of the tire, and returns to the ground exactly one circumference ahead of where it started. That arch is one period of the cycloid. The generating circle has radius r, and each complete revolution produces one arch whose width equals the circumference 2*pi*r and whose height equals the diameter 2r. Cycloids appear in physics, engineering, and architecture. The brachistochrone problem, posed by Johann Bernoulli in 1696, asks: along what curve does a bead slide from one point to another in the shortest time under gravity alone? The answer is the inverted cycloid. The same curve is also the tautochrone: a bead released from any point on an inverted cycloidal ramp takes exactly the same time to reach the bottom regardless of where it starts. Both properties made the cycloid famous as the "Helen of Geometry" in the seventeenth century.
How to use this calculator
Enter the radius of the rolling circle and the rolling angle (the parameter t). Choose whether you prefer degrees or radians for the angle and pick a length unit for all outputs. The calculator immediately returns: - The x and y coordinates of the tracing point at that angle - The arc length of the curve from the start cusp to that angle - The area enclosed between the curve and the baseline up to that angle - The radius of curvature of the curve at that exact point - The five full-arch properties (arc length, area, hump length, height, perimeter) that depend only on the radius Set the rolling angle to 360 degrees (or 2*pi radians) to see the complete single-arch values. Enter angles beyond 360 degrees to explore multi-arch paths: the arc length and enclosed area scale linearly with the number of complete arches.
Cycloid formulas explained
The parametric equations x(t) = r(t - sin t) and y(t) = r(1 - cos t) define the curve, where t is the rolling angle in radians. When t = 0 or t = 2*pi, the point is at a cusp on the baseline, and the curve has zero radius of curvature there (the circle and the line are tangent, and the traced point instantaneously stops). When t = pi (180 degrees), the point is at the apex: x = r*pi and y = 2r. The arc length integral from 0 to t simplifies beautifully: for a single arch, the integrand sqrt(x'(t)^2 + y'(t)^2) = sqrt(2r^2(1-cos t)) = 2r|sin(t/2)|, which integrates to 4r(1-cos(t/2)) from 0 to t, or for a complete arch (t = 2*pi) it gives 8r. Notably, the arc length of one arch equals exactly 8 times the radius, with no pi involved. The area under one arch is computed by integrating y(t)*x'(t) dt from 0 to 2*pi, yielding 3*pi*r^2 (exactly three times the area of the generating circle). The radius of curvature at parameter t is rho = 4r|sin(t/2)|, which ranges from 0 at the cusps to 4r at the apex.
Cycloid variants: curtate and prolate
A standard (or common) cycloid is traced by a point exactly on the rim of the rolling circle. If the tracing point is inside the circle (at distance d < r from the centre), the resulting curve is a curtate cycloid, which never touches the baseline and has no cusps. If the tracing point is outside the circle (d > r), the curve is a prolate cycloid, which crosses itself and forms loops at each period. Both variants are also called trochoids. The epicycloid and hypocycloid are related curves where the rolling circle rolls on the outside or inside of another circle rather than on a straight line. This calculator covers only the standard cycloid traced on a straight baseline.
Key cycloid properties at a glance
| Property | Formula | Relation to circle |
|---|---|---|
| Arc length (one arch) | S = 8r | 8 times the radius |
| Area under one arch | A = 3*pi*r^2 | 3 times the circle area |
| Hump length (cusp span) | C = 2*pi*r | Equal to the circumference |
| Maximum height | d = 2r | Equal to the diameter |
| Full-arch perimeter | p = (8 + 2*pi)*r | ~14.28 times r |
| Radius of curvature | rho = 4r|sin(t/2)| | Max 4r at arch apex |
| x coordinate | x = r(t - sin t) | Parametric (t in radians) |
| y coordinate | y = r(1 - cos t) | Parametric (t in radians) |
All formulas assume a standard cycloid: a point on the rim of a circle of radius r rolling on a flat line without slipping.
Frequently asked questions
Why does one arch of the cycloid have arc length 8r and not a multiple of pi?
The arc length integral for one full arch works out to exactly 8r because the factor of pi from the angle and the factor from the trigonometric substitution cancel perfectly. You can verify this by integrating the speed |v(t)| = 2r|sin(t/2)| from 0 to 2*pi: the antiderivative is -4r*cos(t/2), evaluated from 0 to 2*pi it gives -4r*(-1) - (-4r*1) = 4r + 4r = 8r. This clean result surprised seventeenth-century mathematicians and is one reason the cycloid received so much attention.
What is the brachistochrone and why is it a cycloid?
The brachistochrone (from the Greek for "shortest time") is the curve along which a frictionless bead under gravity descends between two points in the least possible time. Johann Bernoulli proved in 1696 that this curve is an inverted arch of a cycloid. Intuitively, the cycloid front-loads the descent: the curve dips steeply at first, accelerating the bead quickly, then levels out, so it arrives faster than along a straight slope or any other curve.
What is the tautochrone property?
A tautochrone (same-time curve) is a curve along which a bead released from any starting height takes the same time to reach the bottom. Christiaan Huygens showed in 1659 that the inverted cycloid is the unique tautochrone under gravity. He used this to design isochronous pendulum clocks: by constraining the pendulum bob to swing along a cycloidal path, the period becomes independent of the amplitude of swing.
How do I find the x, y position at a given rolling angle?
Convert your angle to radians if needed (degrees x pi/180), then apply the parametric equations: x = r*(t - sin(t)) and y = r*(1 - cos(t)). For example, at 90 degrees (pi/2 radians) with r = 5: x = 5*(pi/2 - sin(pi/2)) = 5*(1.5708 - 1) = 2.854 and y = 5*(1 - cos(pi/2)) = 5*(1 - 0) = 5. This calculator does that arithmetic for you automatically.
What is the radius of curvature of a cycloid and where is it largest?
The radius of curvature rho at parameter t is rho = 4r|sin(t/2)|. At the cusps (t = 0 and t = 2*pi) it is zero, meaning the curve has an infinitely sharp corner there. At the apex (t = pi), sin(pi/2) = 1, so rho reaches its maximum value of 4r: the curve is most gently curved at the top of the arch. At intermediate points the curvature varies smoothly between these extremes.
Is the area under one arch really three times the generating circle?
Yes. Integrating the area element y(t)*x'(t) dt from 0 to 2*pi, substituting the parametric forms, and expanding gives 3*pi*r^2. This is exactly three times the area of the generating circle (pi*r^2). Evangelista Torricelli proved this result around 1644 using early infinitesimal methods, before the development of calculus.
What happens at the cusps of a cycloid?
At a cusp (rolling angle 0, 360 deg, 720 deg, etc.) the tracing point is momentarily in contact with the baseline and has zero velocity relative to the rolling circle. The radius of curvature is zero, meaning the curve meets the line at a point with a vertical tangent on both sides. The curve is continuous but not differentiable at the cusps.