Area of Epicycloid

According to the definition we discussed above, this curve is defined by the parametric equations:

• x(θ)= (R + r) cos(θ)-r cos [Tex](\frac{R + r}{r} \theta)[/Tex]
• y(θ)= (R + r) sin(θ)-r sin [Tex](\frac{R + r}{r} \theta)[/Tex]

where θ is the parameter representing the angle of rotation of the rolling circle.

The area enclosed by an epicycloid can be calculated using the formula:

A=(k+1) (k+2) πr²

Where k is a positive integer.

This formula indicates that the area of the epicycloid is (k+1) (k+2) times larger than the circle traced by the rolling point.

Example of Area of Epicycloid

Suppose we have an epicycloid generated by a fixed circle with radius R=5 units and a rolling circle with radius r=3 units. We want to find the area enclosed by this epicycloid.

Using the formula for the area of an epicycloid when k is a positive integer:

A=(k+1) (k+2)πr²

Substitute the values of k, r, and π into the formula to find the area. Let k=2 for this example.

A= (2+1) (2+2)π(3)²

⇒ A = (3)(4)π(9)

⇒ A = 12⋅9π

⇒ A = 108π

So, the area enclosed by the epicycloid is 108π square units.

Read More,

• Calculus
• Differential Calculus
• Curve Tracing

A curve in geometry known as an epicycloid is created by rolling a point on a circle’s circumference around the exterior of another circle. Designing gears, examining planetary motion, and producing visually arresting patterns are just a few of the disciplines in which epicycloids find usage.

We shall go into depth on the idea of epicycloids in this article. We shall get knowledge about the area, nomenclature, parametric equation, and meaning of epicycloids as well as their characteristics.