The cycloid calculator above works with the curve traced by a marked point on the rim of a circle rolling without slipping along a straight line. Give it the radius and it returns the length of one arch, the width and height of that arch, the area it encloses with the ground, and — at any rolling angle you choose — the exact coordinates, the arc length travelled so far, the speed of the tracing point and the radius of curvature there.
The cycloid is worth knowing because its answers are so unexpectedly clean. Arb Digital publishes this page because the results are exact rather than numerical, and because the curve is the answer to two famous physics problems that most people meet long before they meet the curve itself. For the simpler question of a curved distance along a plain circular arc, our arc length calculator is the right page; this one is about the rolling path, not the rim.
What This Cycloid Calculator Does
It evaluates the standard parametric form of the ordinary cycloid, where the tracing point sits exactly on the rim. The parameter is the rolling angle θ, measured from the instant the point last touched the ground. Since the circle rolls without slipping, that angle is also the distance rolled divided by the radius, so θ carries two meanings at once.
The four headline results describe a single arch. The arch spans one full revolution, which is the ground distance the circle covers before the tracing point comes back down. Its length, base, height and enclosed area are all exact multiples of r or r squared, with no numerical integration involved anywhere.
The angle-dependent results describe one instant of the roll. The tool reports the coordinates of the tracing point, the arc length it has covered along the curve so far, the ratio of its speed to the speed of the circle's centre, the slope of the tangent, and the radius of curvature. Feeding in angles greater than one turn is allowed, and the tool wraps the point into the correct arch while keeping the cumulative arc length running.
Three angle units are offered because all three appear in the literature. Turns are the most intuitive: a quarter turn puts the point a quarter of the way along the arch by angle, though not by arc length, which is one of the details this page is built to make visible.
How to Use It
- Enter the radius of the rolling circle in whatever length unit suits you. Every length output uses the same unit.
- Set the rolling angle and choose degrees, radians or turns. Half a turn is the top of the arch.
- Read the arch figures in the grid: base 2πr, height 2r, area 3πr², with the full arch length 8r in the hero.
- Read the point panel for the coordinates, the speed ratio, the tangent slope and the radius of curvature at your angle.
- Set a number of arches if you want the totals for a wheel making several revolutions along the same line.
The Formula and How It Is Calculated
The parametric equations of the cycloid are
x = r(θ − sin θ) and y = r(1 − cos θ)
with θ in radians. Differentiating gives dx/dθ = r(1 − cos θ) and dy/dθ = r sin θ, so the speed of the tracing point is r√(2 − 2cos θ) = 2r|sin(θ/2)|. Integrating that speed from zero gives the arc length
s(θ) = 4r(1 − cos(θ/2))
which at θ = 2π equals 4r(1 − cos π) = 8r, the length of a whole arch. The area swept under the curve is ∫ y dx = r²∫(1 − cos θ)² dθ, and over a full arch that evaluates to 3πr² — exactly three times the area of the rolling circle. The radius of curvature works out to 4r sin(θ/2), which is twice the instantaneous distance from the point to the ground contact. Wolfram MathWorld's page on the cycloid derives each of these and gives the history of the disputes they caused.
Work the default values through. With r = 2, the arch length is 8 × 2 = 16, the base is 2π × 2 ≈ 12.5664, the height is 4, and the area is 3π × 4 ≈ 37.6991. At θ = 90° = π/2, x = 2(π/2 − 1) ≈ 1.1416 and y = 2(1 − 0) = 2. The arc length so far is 4 × 2 × (1 − cos 45°) = 8(1 − 0.70711) ≈ 2.3431, and the radius of curvature is 4 × 2 × sin 45° ≈ 5.6569.
The Numbers That Surprise People
Three facts about the arch are worth pausing on. The first is that its length, 8r, contains no π whatsoever. A curve built entirely from a rolling circle has a rational length in terms of the radius. Christopher Wren established this in 1658, and it was one of the earliest successful rectifications of a curved line.
The second is that the area under the arch is 3πr², exactly three times the area of the circle that generated it. Not approximately three; exactly. Galileo tried to settle this by cutting the shapes out of sheet metal and weighing them, got a ratio near three, and concluded the true value was probably irrational.
The third is that the arch length 8r is more than twice the ground distance covered by the point's own diameter, and comfortably exceeds the circle's circumference of about 6.28r. The tracing point covers more distance than the hub it belongs to, which is the same reason the top of a rolling wheel moves at twice the vehicle's speed while the contact point is instantaneously stationary. The speed ratio in the point panel shows this directly: it runs from zero at the ground to two at the top.
Why the Cycloid Is the Answer to Two Physics Problems
Turn the arch upside down and it becomes the brachistochrone: the shape of the wire down which a bead, released from rest and moving under gravity alone without friction, reaches a lower point in the least possible time. Not the straight line, which is the shortest path, and not a circular arc. The curve trades a steeper initial drop for speed that is then spent covering ground. MathWorld's page on the brachistochrone problem covers Johann Bernoulli's 1696 challenge and the solutions it drew out.
The same inverted curve is also the tautochrone: a bead released anywhere on it reaches the bottom in the same time, regardless of how high up it started. That is a much stronger statement than the pendulum's small-angle approximation, where the period only becomes amplitude-independent in the limit. Huygens proved it in 1659 and built a pendulum clock with cycloidal cheeks to exploit it. MathWorld's page on the tautochrone problem gives the proof. Our simple pendulum calculator computes the ordinary circular case, where the period does drift with amplitude.
Both results follow from the same fact this page computes: the arc length s = 4r(1 − cos(θ/2)) is a simple trigonometric function of the parameter, which makes the equation of motion along the curve exactly that of a simple harmonic oscillator.
The Cusps, and Why the Curve Is Not Smooth
At θ = 0 and θ = 2π the tracing point is touching the ground, and the curve has a cusp there rather than a smooth minimum. Both derivatives vanish simultaneously, the speed is zero, and the tangent is vertical. The curve arrives and leaves straight down, meeting itself at a sharp point.
This matters in practice. A curve that is continuous but has zero-speed cusps cannot be traversed at constant speed through the cusp, and the curvature there is zero radius — infinitely tight. Any design that follows a true cycloid across the cusp has to handle that discontinuity, which is why real gear and cam profiles use the trochoid family instead, where the tracing point sits inside or outside the rim rather than exactly on it. Our gear ratio calculator handles the transmission side of that geometry.
Between the cusps the curve is perfectly well behaved: smooth, convex, with the maximum at the halfway point where the tangent is horizontal and the speed is exactly twice the centre's. If you only ever work with the interior of an arch, none of the cusp complications arise. The related catenary curve calculator handles the hanging-chain curve, which is often confused with the cycloid because both look like a smooth sag and neither is a parabola.
Arb Digital builds free tools that use closed-form results wherever one exists.
Browse All Free Tools Talk To Our TeamCommon Mistakes to Avoid
- Confusing the arch length with the circumference — the arch is 8r while the circumference is about 6.28r, so the tracing point travels noticeably further than the rim's own perimeter per revolution.
- Assuming a quarter turn is a quarter of the arch — by arc length it is under a sixth, because the point is barely moving near the cusp and fastest near the top.
- Using degrees in the parametric equations — the formulas require radians, and the θ term outside the sine is the one that silently breaks.
- Treating the cycloid as a parabola — it looks like one near the peak but has vertical tangents and cusps at the base, which no parabola has.
- Expecting the curvature to be constant — the radius of curvature runs from zero at the cusps to 4r at the top, so the curve is far tighter at the ends than in the middle.
Related Free Tools From Arb Digital
Measure a plain circular arc with the arc length calculator, work through radius, area and circumference with the circle calculator, time a swing with the simple pendulum calculator, model a hanging chain with the catenary curve calculator, or handle the transmission geometry with the gear ratio calculator. The full free online tools hub lists every geometry tool we publish.
Frequently Asked Questions
It is the path traced by a fixed point on the rim of a circle as that circle rolls without slipping along a straight line. Each revolution produces one arch, meeting the line at a cusp at both ends.
Exactly eight times the radius. The result contains no pi at all, which is why Christopher Wren's rectification of it in 1658 was considered remarkable at the time.
Exactly three times the area of the rolling circle, or three pi r squared. Galileo tried to establish this by weighing metal cut-outs and wrongly concluded the ratio was irrational.
Because an inverted cycloid is the curve down which a bead sliding under gravity without friction gets from one point to a lower one in the least possible time, beating both the straight line and the circular arc.
That a bead released anywhere on an inverted cycloid reaches the bottom in the same time, whatever its starting height. Motion along the curve is exactly simple harmonic, not merely approximately so.
Its speed is twice the centre's speed multiplied by the sine of half the rolling angle. That is zero at the ground contact and exactly double the centre's speed at the top of the arch.
Both derivatives vanish, the speed drops to zero and the tangent becomes vertical. The curve is continuous but not smooth there, and the radius of curvature falls to zero.
No. Those are trochoids, traced by a point inside or outside the rolling circle rather than on it. They have no cusps, and none of the exact results on this page apply to them.
This page explains a classical result in geometry for educational purposes. All figures are exact closed-form values evaluated in double-precision floating point, so the final displayed digits may round.
