🌀 Cycloid Calculator
Calculate properties of a cycloid curve
How to Use This Calculator
Enter Radius
Input the radius of the generating circle that creates the cycloid curve.
Click Calculate
Press the "Calculate Cycloid Properties" button to compute the arc length, area, and period of the cycloid.
Review Results
View the calculated arc length of one arch, the area under one arch, and the period of the cycloid.
Formula
Arc Length (one arch) = 8r
Area (under one arch) = 3πr²
Period = 2Ï€r
Where:
- r = Radius of the generating circle
- π ≈ 3.14159 (Pi constant)
- One arch = one complete cycle of the cycloid
Example: Calculate cycloid properties for r = 5
Arc Length = 8 × 5 = 40 units
Area = 3π × 5² = 3π × 25 = 75π ≈ 235.62 square units
Period = 2π × 5 = 10π ≈ 31.42 units
Parametric Equations:
x = r(θ - sin θ)
y = r(1 - cos θ)
Where θ is the parameter (0 to 2π for one arch)
About Cycloid Calculator
The Cycloid Calculator finds the arc length, area, and period of a cycloid curve. A cycloid is the curve traced by a point on the rim of a circular wheel as it rolls along a straight line without slipping.
When to Use This Calculator
- Mathematics: Study cycloid curves and their properties
- Physics: Analyze brachistochrone problems (fastest path under gravity)
- Engineering: Design cycloidal gears and mechanical systems
- Geometry: Calculate arc lengths and areas of cycloidal curves
- Education: Learn about parametric curves and rolling motion
- Optimization: Study optimal paths in physics and engineering
Why Use Our Calculator?
- ✅ Complete Properties: Calculates arc length, area, and period
- ✅ Instant Results: Get all cycloid properties immediately
- ✅ Step-by-Step Display: See the calculation formulas
- ✅ Educational: Learn cycloid mathematics
- ✅ 100% Accurate: Precise mathematical calculations
- ✅ Completely Free: No registration required
Understanding Cycloids
A cycloid is a remarkable curve with several interesting properties:
- Created by rolling a circle along a straight line
- The arc length of one arch is exactly 8 times the radius (8r)
- The area under one arch is exactly 3 times the area of the generating circle (3πr²)
- One complete arch corresponds to the circle rotating 360° (2π radians)
- Used in the brachistochrone problem: cycloid is the curve of fastest descent
- Has applications in gear design, optics, and optimal path problems
Real-World Applications
Brachistochrone Problem: The cycloid is the solution to the brachistochrone problem - finding the curve of fastest descent between two points under gravity. This was solved by Bernoulli, Newton, and others in the 17th century.
Cycloidal Gears: Cycloidal gear profiles use cycloid curves for smoother, more efficient gear operation with less friction and better load distribution.
Architecture: Cycloidal arches are used in bridge and building design for their structural and aesthetic properties.
Frequently Asked Questions
What is a cycloid?
A cycloid is the curve traced by a point on the rim of a circular wheel as it rolls along a straight line without slipping. It's a parametric curve with unique mathematical properties.
Why is the arc length 8r?
This is a remarkable property of cycloids discovered by calculus. When you integrate the arc length formula for a cycloid from 0 to 2Ï€, the result simplifies to exactly 8 times the radius. This elegant result is independent of calculus integration complexity.
What is the brachistochrone problem?
The brachistochrone problem asks: what is the curve of fastest descent between two points under gravity? The answer is a cycloid. This was one of the most famous problems in calculus of variations, solved by multiple mathematicians in the 17th century.
Can cycloids have different shapes?
Yes! There are variations: hypocycloid (point inside the circle), epicycloid (point outside, rolling on larger circle), and trochoid (point offset from rim). The basic cycloid assumes the point is on the rim.
How is a cycloid different from a sine wave?
A cycloid has sharp cusps (points) at the bottom of each arch, while a sine wave is smooth. Cycloids result from rolling motion; sine waves from circular motion projected onto a line. They have different mathematical forms and properties.
What's the relationship between cycloid and the generating circle?
The cycloid's period equals the circumference of the generating circle (2πr). One complete arch corresponds to one full rotation of the circle. The cycloid's properties (arc length 8r, area 3πr²) are directly related to the circle's radius.
Are cycloids used in real engineering?
Yes! Cycloidal gears use cycloid profiles for efficient, low-friction gear operation. Cycloidal motion is also used in cam designs, pumps, and mechanical systems where smooth, predictable motion is needed.