Editing Cycloid (section)

== Arc length ==
[[File:Cycloid length.png|thumb|The length of the cycloid as consequence of the property of its involute]]
The [[arc length]] {{mvar|S}} of one arch is given by
<math display="block">\begin{align}
  S &= \int_0^{2\pi} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt \\
    &= \int_0^{2\pi} r \sqrt{2 - 2\cos t}\, dt \\
    &= 2r\int_0^{2\pi} \sin \frac{t}{2}\, dt \\
    &= 8r.
 \end{align}</math>

Another geometric way to calculate the length of the cycloid is to notice that when a wire describing an [[#Involute|involute]] has been completely unwrapped from half an arch, it extends itself along two diameters, a length of {{math|4''r''}}. This is thus equal to half the length of arch, and that of a complete arch is {{math|8''r''}}.

From the cycloid's vertex (the point with a horizontal tangent and <math>\cos t=-1</math>) to any point within the same arch, the arc length squared is <math>8r^2(1+\cos t)</math>, which is proportional to the height difference <math>r(1+\cos t)</math>; this property is the basis for the cycloid's [[Tautochrone curve#Lagrangian solution|isochronism]]. In fact, the arc length squared is equal to the height difference multiplied by the full arch length {{math|8''r''}}.