Cycloid Calculator
Calculate properties of a cycloid with support for both degrees and radians angle inputs
Category: Mathematics
Cycloid Calculator Inputs
Cycloid Calculator Formula
Equation
x = r(θ - \sinθ), y = r(1 - \cosθ)
Excel Formula
=x=r(-),y=r(1-)
Variables
- Radius (r) — Enter the Radius (r) value used by the Cycloid Calculator.
- Angle (θ) — Enter the Angle (θ) value used by the Cycloid Calculator.
- Angle Unit — Select the unit for the angle input
How the Cycloid Calculator Works
A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slipping. This elegant curve has fascinated mathematicians for centuries and appears in numerous applications from gear design to optimal path problems.
The core relationship is x = r(\theta - \sin\theta), y = r(1 - \cos\theta). Typical inputs include Radius (r), Angle (θ), Angle Unit.
Enter your values in the cycloid calculator above, review the step-by-step solution, and compare against the worked examples below so you can see how each input changes the result. This free online mathematics tool is built for homework, design checks, and professional verification.
Cycloid Calculator Theory & Explanation
Parametric Equations and Derivation
The cycloid is defined by the parametric equations: x = r(θ - \sinθ), y = r(1 - \cosθ), where r is the radius of the generating circle and θ is the angle of rotation in radians. These equations can be derived by considering the motion of a point on a rolling circle. When a circle of radius r rolls without slipping, the center moves horizontally at a distance rθ, while the point on the rim traces a path that combines this horizontal motion with the circular motion around the center. For angles in degrees, convert using θ_rad = θ_deg × (π)/(180).
x = r(θ - \sinθ), \quad y = r(1 - \cosθ) \quad \text(where θ \text is in radians)
Geometric Properties
The cycloid has several remarkable geometric properties: 1) It is a periodic curve with period 2π r in the x-direction, 2) Each arch has a maximum height of 2r, 3) The curve has cusps (sharp points) at θ = 2nπ radians (or θ = 360n°) where it touches the ground, 4) The curve is symmetric about the vertical line through its highest point, 5) The tangent at any point makes an angle θ/2 with the horizontal (in radians) or θ/2 × 180/π degrees.
\textPeriod = 2π r, \quad \textMax height = 2r, \quad \textTangent angle = (θ)/(2) \text (radians) = (θ)/(2) × (180°)/(π)
Arc Length and Curvature
The arc length of one complete arch of a cycloid is 8r, which is four times the diameter of the generating circle. This remarkable property was first proven by Christopher Wren. The curvature of the cycloid is given by \kappa = (1)/(4r\sin(θ/2)), showing that the curvature is infinite at the cusps and decreases as we move away from them. The total curvature over one arch is π radians (180°). For angles in degrees, use \sin(θ/2) where θ is converted to radians.
L = 8r, \quad \kappa = (1)/(4r\sin(θ/2)), \quad ∫_0^2π \kappa \, ds = π \text radians = 180°
Area Under One Arch
The area under one complete arch of a cycloid is 3π r^2, which is three times the area of the generating circle. This can be calculated using the parametric integral: A = ∫_0^2π y (dx)/(dθ) dθ = ∫_0^2π r(1-\cosθ) · r(1-\cosθ) dθ = 3π r^2. This result was also proven by Wren and is known as Wren's theorem. The integration is performed over θ from 0 to 2π radians (0° to 360°).
A = 3π r^2 = 3 × π r^2, \quad \textintegrated from θ = 0 \text to 2π \text radians (0° to 360°)
Involute and Evolute
The involute of a cycloid is another cycloid of the same size, but shifted. This means that if you "unwind" a string from a cycloid, the end of the string traces out another cycloid. The evolute (the locus of centers of curvature) of a cycloid is also a cycloid, but scaled and shifted. This property makes cycloids useful in gear design.
\textInvolute of cycloid = \textAnother cycloid
Brachistochrone Property
The cycloid is the solution to the brachistochrone problem: among all curves connecting two points, the cycloid is the one along which a particle slides under gravity in the shortest time. This was proven by Johann Bernoulli in 1696. The time of descent is T = π√(\fraca)g, where a is the radius of the generating circle and g is gravitational acceleration. The angle parameter θ ranges from 0 to π radians (0° to 180°) for the complete brachistochrone path.
T = π√(\fraca)g \quad \text(brachistochrone time), \quad θ \in [0, π] \text radians (0° to 180°)
Tautochrone Property
The cycloid is also a tautochrone: regardless of where a particle starts on an inverted cycloid, it takes the same time to reach the bottom. This property was used by Christiaan Huygens to design the cycloidal pendulum, which has a constant period independent of amplitude. The angle range for the tautochrone is θ \in [0, π] radians (0° to 180°).
T = π√(\fraca)g \quad \text(constant for all starting positions), \quad θ \in [0, π] \text radians (0° to 180°)
Mathematical Relationships
Several important relationships exist: 1) The slope of the tangent is (dy)/(dx) = (\sinθ)/(1-\cosθ) = \cot(θ/2), 2) The radius of curvature is \rho = 4r\sin(θ/2), 3) The cycloid can be expressed in Cartesian coordinates as x = r\cos^-1(1-y/r) - √(2ry-y^2), 4) The area moment of inertia about the x-axis is I_x = (5π r^4)/(4). All angle calculations use radians; for degrees, convert using θ_rad = θ_deg × (π)/(180).
(dy)/(dx) = \cot(θ)/(2), \quad \rho = 4r\sin(θ)/(2), \quad I_x = (5π r^4)/(4) \quad \text(all angles in radians)
Cycloid Calculator Worked Examples
Worked Example
Inputs
- radius: 2
- angle: 90
- angleUnit: degrees
Result: x = 1.14, y = 2.00, arcLength = 16.00, areaUnderArch = 37.70, angleInRadians = 1.57 rad
Explanation
For r = 2 and θ = 90°: x = 2(π/2 - sin(π/2)) = 2(1.57 - 1) = 1.14, y = 2(1 - cos(π/2)) = 2(1 - 0) = 2. The arc length of one complete arch is 8r = 16, and the area under one arch is 3πr² = 37.70. The angle in radians is π/2 ≈ 1.57 rad. The curvature at this point is κ = 1/(4r sin(θ/2)) = 1/(8 sin(45°)) ≈ 0.18.
Second Scenario
Inputs
- radius: 1.5
- angle: 90
- angleUnit: degrees
Result: x = 1.14, y = 2.00, arcLength = 16.00, areaUnderArch = 37.70, angleInRadians = 1.57 rad
Explanation
This scenario uses different inputs (radius = 1.5, angle = 90, angleUnit = degrees) to show how changing one variable affects the cycloid result. Run the calculator above with these values to get the exact updated output with step-by-step work.
Common Cycloid Calculator Use Cases
- Homework and exam practice
- Engineering and science coursework
- Quick verification of hand calculations
- Cycloid homework and study
- Cycloid design and analysis
Cycloid Calculator FAQs
What is a cycloid used for?
Cycloids are used in gear design, pendulum clocks (cycloidal pendulum), and various mechanical applications where smooth rolling motion is required. They are also used in cam profiles, gear teeth design, and as the optimal path for gravity-driven systems.
What is the brachistochrone problem?
The brachistochrone problem asks for the curve between two points that allows a particle to fall under gravity in the shortest time. The solution is a cycloid, proven by Johann Bernoulli in 1696. This property makes cycloids important in physics and engineering applications.
How does the cycloid relate to the circle?
The cycloid is generated by a point on a rolling circle. The arc length of one arch is 8r (four times the diameter) and the area under one arch is 3πr² (three times the circle area). The cycloid's involute and evolute are also cycloids, making it self-similar in certain transformations.
What are the applications of cycloids in physics?
Cycloids appear in the study of rolling motion, pendulum dynamics, and optimal path problems. They are also used in the design of cam profiles, gear teeth, and as the solution to the tautochrone problem where particles take the same time to reach the bottom regardless of starting position.
What is the tautochrone property?
The tautochrone property means that regardless of where a particle starts on an inverted cycloid, it takes the same time to reach the bottom. This property was used by Christiaan Huygens to design the cycloidal pendulum, which maintains constant period independent of amplitude.
How is the cycloid curvature calculated?
The curvature of a cycloid is given by κ = 1/(4r sin(θ/2)). This shows that the curvature is infinite at the cusps (where θ = 2nπ) and decreases as we move away from them. The total curvature over one complete arch is π.
What is Wren's theorem?
Wren's theorem states that the area under one complete arch of a cycloid is 3πr², which is three times the area of the generating circle. This was proven by Christopher Wren, who also proved that the arc length of one arch is 8r.
How do cycloids relate to gear design?
Cycloids are used in gear design because their involute and evolute properties create smooth, efficient gear tooth profiles. The cycloidal gear tooth shape provides better load distribution and reduced wear compared to traditional involute gears in certain applications.
What is the mathematical significance of the cycloid?
The cycloid is significant because it solves multiple optimization problems (brachistochrone, tautochrone), has remarkable geometric properties (self-similarity, constant curvature relationships), and appears naturally in rolling motion. It represents one of the first examples of calculus of variations and optimal control theory.
How do you derive the parametric equations?
The parametric equations x = r(θ - sin θ), y = r(1 - cos θ) are derived by considering a circle of radius r rolling without slipping. The center moves horizontally by rθ, while a point on the rim has coordinates (r cos θ, r sin θ) relative to the center. Combining these gives the cycloid equations.
Can I input angles in both degrees and radians?
Yes! The calculator supports both degrees and radians. You can select your preferred angle unit from the dropdown menu. The calculator will automatically convert between units and display both the angle in degrees and radians in the results for your convenience.