A cycloid is the path traced by a point on a circle’s rim as the circle rolls along a straight line without slipping. One turn produces one arch between cusps. This calculator uses the radius to report area under one arch, curved length, height, horizontal width and perimeter.
The geometry assumes an ideal circle, flat baseline and pure rolling. Sliding, or moving the point away from the rim, produces a different curve. Linear results use the radius unit; area uses its squared unit.
The curve can be parameterized by the circle’s rotation angle t in radians: x = r(t − sin t) and y = r(1 − cos t). As t runs from 0 to 2π, the circle makes one revolution and the tracing point moves from one cusp to the next. The calculator does not ask for t because it reports properties of that entire arch rather than coordinates at a selected point.
The page labels this straight cusp-to-cusp result “Horizontal arch width,” which distinguishes it from the curved arc length.
For one arch generated by a circle of radius r:
The parametric equations are x = r(t − sin t) and y = r(1 − cos t), for 0 ≤ t ≤ 2π. The perimeter here includes both the cycloid arc and the straight baseline from one cusp to the next.
| Symbol or result | Meaning |
|---|---|
| r | Radius of the rolling circle, measured from its centre to the rim. |
| t | Rotation parameter in radians; one full arch corresponds to 0 through 2π. |
| A | Area bounded by one cycloid arch and the straight baseline, in squared units. |
| S | Length measured along the curved cycloid from one cusp to the next. |
| H | Maximum height of the arch above the baseline, equal to the rolling circle’s diameter. |
| W | Horizontal cusp-to-cusp distance, shown by the page as “Horizontal arch width.” |
| P | Boundary length of the region under one arch: curved arc plus straight baseline. |
Let the rolling circle have radius 2.5 cm.
These match the calculator output, including the selected square-centimetre area unit. The numerical value 58.90 is not automatically converted between unit systems.
Linear dimensions scale directly with r. Doubling the radius doubles arc length, height, width and perimeter. Area scales with r², so doubling the radius quadruples area.
Arc length and width describe different paths. The 8r arc follows the curve. The 2πr width lies on the baseline and equals the distance rolled in one no-slip revolution. Their sum is the perimeter of the region that includes the baseline.
The unit selector labels the input; it does not convert an existing number. Entering 2.5 after switching from centimetres to metres means 2.5 metres. Apply the squared unit to area. The calculator rounds to two decimal places.
A negative radius has no ordinary geometric meaning, and zero produces a degenerate arch. Use a finite radius greater than zero.
It is the horizontal width from one cusp to the next, 2πr. The distance along the curved cycloid is a separate output, the arc length 8r.
At the top of the arch, the tracing point is one radius above the circle’s centre and the centre is one radius above the baseline, giving a total height of 2r.
Use the square of the selected radius unit: mm², cm², m², in² or ft². The calculator displays that squared unit with the area.
No. The same typed number is interpreted in the newly selected unit. Convert the numerical radius yourself if you want to preserve the same physical size.
One arch takes one full turn. With no slipping, the horizontal distance rolled equals the circle’s circumference, 2πr.
No. This page’s perimeter encloses the region under one arch, so it adds the curved length 8r and the straight baseline width 2πr.
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