Cycloid Calculator is used for calculating every aspect of a cycloid, including its perimeter, area, arc length of a cycloid, hump length, hump height and more. Just by entering one of the parameters, the calculator uses the equation formulas to calculate all other parts of a cycloid based on the input values. You can choose in which square unit you want to calculate the cycloid, either millimetres, centimetres, meters, inches, feet or other less common square units. Instead of using the formulas and calculating aspects of cycloids manually, you can leave all of that to our online calculator.

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What is a cycloid? Cycloid definition

A cycloid is a curve that rolls along a particular line, leaving traces behind, which look like a few half circles with specific radii R. Cycloid is an even linear and circular motion with a constant speed.

In 1599, it was the first time when a cycloid was mentioned and used, when Galileo cut pieces of metal into the cycloidal shapes, which he used trying to find the area. Cycloid found its fame after Melville’s book “Moby Dick” and the important passage where it is claimed, “in geometry all bodies gliding along a cycloid”. In the 17th Century, many mathematicians began exploring more and more about cycloids. As a result, they gave it a unique name, “Helen of Geometers”. The circle that is moving is called a generating circle, while the line along which it rolls is called a base line.

Cycloid equation

Cycloid illustration

If we take a look at the picture above. We see a circle rolling along the path on the positive side of a cartesian coordinate plane. Let’s consider a point P on the circle’s circumference, which has its coordinates P (x,y). This point is at the centre of the cartesian plane when the circle is not moving and aligned to the y – axis. When the circle starts rolling along the straight line, we can observe that the point P and its coordinates change since the circle is in movement.

We can define the equation of the point P (x,y) coordinates as follows:

Polar equation

Parametric equations:

x = r (\theta - sin\theta)

y = r (1 - cos\theta )

r – radius of the circle

θ – an angle at which the circle is moving

Cartesian equation

x = r \times cos ^{-1} (1 - \frac{y}{r}) - \sqrt{y \times (2r - y)}

Arc length of a cycloid equation

S = 8 \times r

S – the distance between two cusps

r – radius of the circle

Area of a cycloid equation

A = 3 \times \pi \times r^{2}

A – the area under the cycloidal arch that encloses the space with an x-axis line.

r – circle radius

Hump length equation

C = 2 \times \pi \times r

C – the distance between two cusps, often called the circumference

r – circle radius

Hump height equation

d = 2 \times r

d – the distance between the x-axis line and one point on the rolling circle (diameter)

r – circle radius

The perimeter of a cycloid equation

P = C + S

P – General definition of a perimeter is the sum of all sides of a particular shape. Since a cycloid has 2 sides: the arch and the base, we can calculate the perimeter by adding those two sides.

C – arc length of a cycloid

S – cycloid’s base

Cycloid curves

The way cycloids roll along the straight line is through a constant pattern leaving the same curve with the same speed from the first revolution until the last it makes.

Cycloid example

Tire moving on the road same as a cycloid rolling on the straight line

Where do we use the concept of cycloids? First, let’s observe the picture above and tell me what you see. A car, a tire moving, right? Tires work the same way as cycloids because they leave traces of these cycloid curves where the vehicle is moving. But, of course, we need to assume that the car is moving at a constant speed. Otherwise, this scenario cannot be used as an example of cycloids and their use in real-life.

Variations of a cycloid

There are a few known types of cycloids in math. So, let’s cover them briefly in this section:

Curtate cycloid

A curtate cycloid is a cycloid that is made when a point, anywhere but not in the centre of a circle, rolls along the line and leaves a trace of curves behind.

Prolate cycloid

A prolate cycloid is a cycloid without cusps and does not intersect the base line.

Hypocycloid

Hypocycloid is a curve made by the traces of a small circle rolling without slipping inside a bigger (base) circle.

Epicycloid

Epicycloid is a plane curve made by the traces of a circle rolling around a bigger (fixed) circle.

Hypotrochoid

A hypotrochoid is a curve made by the traces of a point outside of a circle rolling inside of another bigger circle.

Epitrochoid

Epitrochoid is a curve made by the traces of a point inside one circle (not in the centre) rolling around another circle.

What is the diameter of a cycloid?

The diameter of a cycloid is equal to the height of a cycloid’s arc. It is the distance between the base line and one point on the cycloid’s curve. It is dependent on the radius of the circle. The bigger radius is, the larger diameter a cycloid has.

How to calculate the diameter of a cycloid?

In order to calculate the diameter of a cycloid, you need to know the radius of a circle that rolls along the line in the cartesian system. The diameter grows proportionally with the rise of the radius. Let’s take a look at the formula and use it in the example below. We will use our Cycloid Calculator to find the diameter.

Formula: D = 2 \times r

Example: Let’s assume we have a rolling circle of a radius of 30 cm. The circle goes rolling along the path and leaves traces of a curve behind. Now, every curve it makes has the same diameter. So how do we calculate the diameter of a cycloid? We take the radius length and multiply it by 2.

Diameter of a cycloid = 2 \times 30 = 60 cm