Cycloid Calculator
Calculate cycloid properties including area, arc length, and parametric coordinates
Calculate Cycloid Properties
Radius of the circle that rolls along the line
Angular parameter for calculating specific point coordinates (0 to 2π for one arch)
Cycloid Properties
Key Formulas
Area: A = 3πr²
Arc Length: S = 8r
Hump Length: C = 2πr
Hump Height: d = 2r
Perimeter: p = S + C
Parametric Equations
x = r(t - sin t)
y = r(1 - cos t)
where t is the angular parameter
One complete arch: t ∈ [0, 2π]
Example Calculation
Rolling Coin Example
Circle radius: 2.5 cm (diameter = 5 cm coin)
Problem: Find the area under one arch of the cycloid
Given: r = 2.5 cm
Solution Steps
1. Area formula: A = 3πr²
2. Substitute: A = 3 × π × (2.5)²
3. Calculate: A = 3 × π × 6.25
4. Result: A = 18.75π ≈ 58.91 cm²
Additional Properties
Arc length: S = 8 × 2.5 = 20 cm
Hump length: C = 2π × 2.5 ≈ 15.71 cm
Hump height: d = 2 × 2.5 = 5 cm
Perimeter: p = 20 + 15.71 = 35.71 cm
Cycloid Components
Area
A = 3πr²
Area under one arch
Arc Length
S = 8r
Curved distance between cusps
Hump Length
C = 2πr
Base length (circumference)
Hump Height
d = 2r
Maximum height (diameter)
Cycloid Variations
Standard Cycloid
Point on circle circumference
Curtate Cycloid
Point inside the circle
Prolate Cycloid
Point outside the circle
Epicycloid
Circle rolling around outside
Hypocycloid
Circle rolling around inside
Understanding Cycloids
What is a Cycloid?
A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line. This fascinating curve has captured mathematicians' attention for centuries due to its unique properties and practical applications.
Historical Significance
- •Galileo coined the term "cycloid" and studied its properties
- •Known as the "Helen of geometry" due to disputes over its discovery
- •Brachistochrone problem: fastest descent curve
- •Tautochrone problem: equal time descent
Mathematical Properties
Parametric Equations
x = r(t - sin t)
y = r(1 - cos t)
Remarkable Properties
Area is exactly 3 times the generating circle's area
Arc Length
Exactly 8 times the generating circle's radius
Applications: Gear design, pendulum clocks, architectural arches, and optimal curve problems in physics and engineering.