Cycloid Calculator

Your details

Choose whether to enter the rolling angle in degrees or radians.
The unit for the radius and all length outputs.
Radius of the rolling circle. All length outputs are in the same unit.
cm
The angle the circle has rolled. 360 deg (2 pi rad) = one complete arch. Use values above 360 deg for multi-arch paths.
deg
y coordinate
10

Vertical position of the tracing point: y = r(1 - cos t)

x coordinate15.708
Arc length to angle t20
Enclosed area to angle t78.5398
Radius of curvature20
Full-arch arc length (S)40
Full-arch area (A)235.6194
Hump length (C)31.4159
Height (d)10
Full-arch perimeter (p)71.4159
Length unitcm
Arc length to t20
Full-arch arc length40
Hump length (C)31.4159
Height (d)10

The rolling circle is 50% through its first complete arch.

  • At t = 180.0 deg the tracing point is at (15.708, 10.000) cm.
  • The curve has traveled 20.0000 cm so far, which is 50.0% of a full arch (40.0000 cm).
  • The radius of curvature at this angle is 20.0000 cm (the osculating circle that best approximates the curve locally has this radius).
  • One full arch encloses an area of 235.6194 cm^2 (exactly 3 times the area of the generating circle, pi*r^2 = 78.5398 cm^2).

Next stepSet the rolling angle to 360 degrees to see the full single-arch properties, or try different radii to scale the curve.

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