The formula for the area of a circular path, often referred to as an annulus, is calculated by subtracting the area of the inner circle from the area of the outer circle.
The exact formula is:
A = π(R² - r²)
Where:
- A represents the area of the circular path.
- π (Pi) is a mathematical constant approximately equal to 3.14159.
- R is the radius of the outer circle.
- r is the radius of the inner circle.
Understanding the Circular Path (Annulus)
A circular path, or annulus, is the region between two concentric circles (circles that share the same center but have different radii). Imagine a ring or a washer; its flat surface represents an annulus.
To find the area of this specific shape, we rely on the fundamental formula for the area of a single circle. As noted, the area of a circle is calculated by multiplying pi (π) by the square of its radius. Specifically, the area of a circle when the radius 'r' is given is πr².
Deriving the Formula
To derive the formula for a circular path, follow these simple steps:
- Calculate the area of the outer circle: Using the larger radius (R), the area of the outer circle (A_outer) is πR².
- Calculate the area of the inner circle: Using the smaller radius (r), the area of the inner circle (A_inner) is πr².
- Subtract the inner area from the outer area: The area of the circular path (A) is the difference:
A = A_outer - A_inner
A = πR² - πr²
A = π(R² - r²) (by factoring out π)
This method effectively removes the central, unpathed circular region, leaving only the area of the path itself.
Key Components of the Formula
| Component | Description |
|---|---|
| A | Total area of the circular path (annulus). |
| π (Pi) | Mathematical constant (approx. 3.14159). |
| R | Radius of the outer circle (from center to outer edge). |
| r | Radius of the inner circle (from center to inner edge). |
For a deeper understanding of circle geometry, you can explore resources like Wikipedia's article on Annulus or Khan Academy's lessons on area of circles.
Practical Example
Let's calculate the area of a circular path with the following dimensions:
- Outer radius (R) = 10 meters
- Inner radius (r) = 6 meters
- Calculate R²: 10² = 100
- Calculate r²: 6² = 36
- Find the difference R² - r²: 100 - 36 = 64
- Multiply by π: A = π * 64
So, the area of the circular path is 64π square meters.
If you need a numerical approximation, using π ≈ 3.14159:
A ≈ 64 * 3.14159 ≈ 201.06 square meters.
This formula is widely used in engineering, architecture, and design for calculating the area of components like washers, pipes, and circular tracks.
