Calculating the perimeter of an ellipse can be accurately approximated using an alternative Ramanujan formula. This article will guide you through the process using the formula
\[ P = \pi \cdot (a + b) \cdot \left(1 + \dfrac{3 \cdot h}{10 + \sqrt{4 - 3 \cdot h}}\right) \]
where
\[ h = \dfrac{(a - b)^2}{(a + b)^2} \]
We will explain the formula and provide a step-by-step example to illustrate the calculations.
The Formula for the Perimeter of an Ellipse
The perimeter \( P \) of an ellipse is approximated by:
\[ P = \pi \cdot (a + b) \cdot \left(1 + \dfrac{3 \cdot h}{10 + \sqrt{4 - 3 \cdot h}}\right) \]
where
\[ h = \dfrac{(a - b)^2}{(a + b)^2} \]
Explanation of the Formula
1. **\( \pi \cdot (a + b) \)**: This term is the sum of the semi-major axis \( a \) and the semi-minor axis \( b \), multiplied by Pi (\( \pi \)).
2. **\( h = \dfrac{(a - b)^2}{(a + b)^2} \)**: This term calculates \( h \) as the square of the difference between the semi-major and semi-minor axes divided by the square of their sum.
3. **\( 1 + \dfrac{3 \cdot h}{10 + \sqrt{4 - 3 \cdot h}} \)**: This term adjusts the result based on the value of \( h \) to provide a more accurate approximation of the perimeter.
Step-by-Step Calculation
Let's work through an example to illustrate the process.
Example:
Suppose we have an ellipse with a semi-major axis \( a = 10 \) units and a semi-minor axis \( b = 6 \) units. We want to find the perimeter of the ellipse.
Step 1: Identify the Given Values
Given:
- Semi-major axis \( a = 10 \) units
- Semi-minor axis \( b = 6 \) units
Step 2: Calculate \( h \)
\[ h = \dfrac{(10 - 6)^2}{(10 + 6)^2} = \dfrac{4^2}{16^2} = \dfrac{16}{256} = 0.0625 \]
Step 3: Substitute \( h \) into the Formula
\[ P = \pi \cdot (10 + 6) \cdot \left(1 + \dfrac{3 \cdot 0.0625}{10 + \sqrt{4 - 3 \cdot 0.0625}}\right) \]
Step 4: Calculate the Perimeter
First, calculate the inner terms:
\[ 3 \cdot 0.0625 = 0.1875 \]
\[ 4 - 3 \cdot 0.0625 = 4 - 0.1875 = 3.8125 \]
\[ \sqrt{3.8125} \approx 1.952 \]
Now, substitute these values back into the formula:
\[ 10 + 1.952 = 11.952 \]
\[ 1 + \dfrac{0.1875}{11.952} \approx 1 + 0.0157 = 1.0157 \]
\[ P = \pi \cdot 16 \cdot 1.0157 \]
Finally, multiply by \( \pi \) (approximated as 3.14159):
\[ P = 3.14159 \cdot 16 \cdot 1.0157 \]
\[ P \approx 51.067 \]
Final Value
For an ellipse with a semi-major axis of 10 units and a semi-minor axis of 6 units, the approximate perimeter is 51.07 units.
This method using an alternative Ramanujan formula provides a highly accurate approximation for the perimeter of an ellipse, making it practical for various applications.
