Annulus Formula | Learn with solved examples

by Yogi P - July 14, 2024

What is the Annulus Formula?

Annulus formula is a mathematical expression used for finding area of annulus which are formed between two concentric circles. An annulus can be defined as a flat shape that is enclosed by the circumferences of two concentric circles with one inside the other. For example, an iris of an eye or a disk of an astronomical object.

Geometry refers to annulus as 2D ring-like figures with different radius bounded by two concentric circles; this name has its origin from little ring Latin word. This particular geometric shape is common in different fields such as engineering, architecture and mathematics. To calculate the area of this kind of shape so let’s know about the annulus formula.

Annulus Definition

An annulus is defined by having two radii and these are represented by capital letter R (denotes larger circle) and small case r(denotes smaller circle) whereas R > r. In addition, it creates a space between two circles known as annuli.

Calculating Annular Area

The difference between bigger and smaller circle areas gives the area of an annulus, which in turn can be calculated using formula for finding circle area:

A = 𝜋r² , where r is the radius of the circle.

Derivation of the Annulus Formula

To find the area of the annulus, we subtract the area of the smaller circle from the area of the bigger circle:

Area of the bigger circle = 𝜋R²

Area of the smaller circle = 𝜋r²

Therefore, the area of the annulus A is given by:

A = 𝜋R² − πr²

This formula can be factored to make it more compact:

A = 𝜋( R² − r² )

Application of the Annulus Formula

The annulus formula is used in various practical applications:

• Engineering: In mechanical engineering, annuli are often encountered in the design of components like washers, gaskets, and bearings.
• Architecture: Annular shapes are common in architectural designs, particularly in the creation of decorative elements and structural components.
• Physics: In physics, annuli can describe regions of material with specific properties, such as magnetic fields or fluid flow in pipes.

Example Calculation

Let’s consider an example to illustrate the application of the annulus formula. Suppose we have an annulus with an outer radius R = 10 units and an inner radius r = 6 units. To find the area of this annulus, we use the formula:

A = 𝜋( R2 − r2 )

First, calculate the squares of the radii:

𝑅² = 10² = 100

r² = 6² = 36

Next, subtract the area of the smaller circle from the area of the larger circle:

R² − r² = 100 − 36 = 64

Finally, multiply by 𝜋:

A = 𝜋×64

Using
𝜋 ≈ 3.14159 , the area is approximately:

A ≈ 3.14159×64 ≈ 201.06 square units

Use Cases

The Annulus Formula is often used to calculate areas in engineering and other practical applications, such as calculation of gear sizes for machines, area of beams and support structures, and radius of bumps or ridges in engineering products. It can also be used in problems related to volume, surface area, and perimeter calculations.

Conclusion
The annulus formula A = 𝜋( R2 − r2 ) is a straightforward yet powerful tool for calculating the area of an annular region. This geometric shape appears in numerous real-world contexts, making the formula valuable across various subjects. Knowing how to derive and apply this formula enhances one’s ability to solve practical problems involving annular shapes.