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Cycloid

A cycloid is the curve traced by a point on a circle rolling along a line. Arc length per arch = 8r (no π!); area per arch = 3πr². Solves the brachistochrone.

Concept Fundamentals
L = 8r per arch
Arc length
A = 3πr² per arch
Area
x=r(t−sin t), y=r(1−cos t)
Parametric
Fastest descent curve
Brachistochrone

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Arc length 8r has no π—unusual for a curve involving circles. The cycloid solves the brachistochrone: fastest path under gravity. Huygens used cycloidal cheeks for accurate pendulum clocks.

Key quantities
L = 8r per arch
Arc length
Key relation
A = 3πr² per arch
Area
Key relation
x=r(t−sin t), y=r(1−cos t)
Parametric
Key relation
Fastest descent curve
Brachistochrone
Key relation

Ready to run the numbers?

Why: Cycloids appear in gear teeth, Ferris wheels, and roller coasters. The brachistochrone (fastest descent) is a cycloid arc.

How: Arc length = 8r per arch (elegant—no π). Area = 3πr² per arch—exactly 3× the rolling circle. Parametric: x=r(t−sin t), y=r(1−cos t).

Arc length 8r has no π—unusual for a curve involving circles.The cycloid solves the brachistochrone: fastest path under gravity.
Sources:Wolfram MathWorld - CycloidWikipedia - Cycloid

Run the calculator when you are ready.

Cycloid CalculatorEnter radius and number of arches to compute arc length and area
🌀
PARAMETRIC CURVERolling Circle

Cycloid — Path of a Rolling Point

Arc length = 8r per arch. Area = 3πr² per arch. The curve traced by a point on a rolling circle.

Circle →Arc Length →

🌀 Examples — Click to Load

Calculation Settings

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

🌊

Cycloid arc length = 8r per arch—no π in the formula!

— Formula

⚡

The brachistochrone (fastest descent) is a cycloid arc.

— Physics

📋 Key Takeaways

  • • A cycloid is the curve traced by a point on a circle rolling along a line
  • • Arc length of one arch = 8r (elegant formula — no π!)
  • • Area under one arch = 3πr² — exactly 3× the area of the rolling circle
  • • The cycloid solves the brachistochrone (fastest descent) and tautochrone problems
  • • Huygens used cycloidal cheeks for accurate pendulum clocks

💡 Did You Know?

🎡A point on a Ferris wheel traces a cycloid as the wheel rolls. With r=25m, one arch is 200m long!Source: Physics
⚙️Cycloidal gears have teeth shaped like cycloids — they mesh smoothly with less wearSource: Mechanical Engineering
🕰️Christiaan Huygens (1659) used cycloidal pendulum cheeks so the bob follows a cycloid — constant periodSource: Horology
🎢The brachistochrone: a cycloid is the curve of fastest descent between two points under gravitySource: Calculus of Variations
🪙Roll a coin on a table — a point on its edge traces a cycloid (or curtate if inside the edge)Source: Geometry
🌀Cycloids appear in animation for smooth rolling motion paths and spirograph-style designsSource: Computer Graphics

📖 Cycloid Formulas

Parametric Equations

x=r(t−sin⁡t),y=r(1−cos⁡t)x = r(t - \sin t), \quad y = r(1 - \cos t)

t is the angle (radians) the circle has rotated. One arch: t = 0 to 2π.

Arc Length & Area

Larch=8r,Aarch=3πr2L_{\text{arch}} = 8r, \quad A_{\text{arch}} = 3\pi r^2

For n arches: L = 8rn, A = 3πr²n.

🎯 Expert Tips

Quick Arc Length

One arch = 8r. So r=10m → 80m per arch. No π needed!

Area vs Circle

Area under one arch = 3πr² = 3× the rolling circle area. Memorable!

Multiple Arches

For n complete arches, multiply by n: L=8rn, A=3πr²n.

Brachistochrone

The cycloid is the curve of fastest descent — a bead slides fastest along it.

⚖️ Comparison

PropertyCycloid ArchCircle
Arc length8r2πr
Area3πr²πr²
Ratio4/π ≈ 1.271

📊 Quick Facts

8r
Arc per Arch
3πr²
Area per Arch
2π
One Arch (rad)
3×
vs Circle Area

❓ FAQ

What is a cycloid?

The curve traced by a point on a circle as it rolls along a straight line without slipping.

Why is arc length 8r?

Derived from the parametric equations. The integral ∫√(dx²+dy²) from 0 to 2π equals 8r.

Why is area 3πr²?

The area under one arch equals 3× the area of the rolling circle. Integral of y dx gives 3πr².

What is the brachistochrone?

The curve of fastest descent under gravity. Johann Bernoulli proved it's a cycloid (1696).

What is the tautochrone?

A curve where descent time is independent of starting point. The cycloid has this property.

How do pendulum clocks use cycloids?

Huygens placed cycloidal "cheeks" so the pendulum bob follows a cycloid — constant period.

Curtate vs prolate?

Curtate: point inside the circle. Prolate: point outside. This calculator uses standard (point on circumference).

Units?

Use any length unit: m, cm, in, px. Results match your unit.

⚠️ Disclaimer: This calculator uses the standard cycloid (point on circumference). Curtate and prolate cycloids have different formulas. Results are for ideal mathematical curves.

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