About Ellipse Perimeter Calculation

What is the Perimeter of an Ellipse?

The perimeter of an ellipse, also known as its circumference, is the distance around the ellipse. Unlike a circle, an ellipse's perimeter doesn't have a simple exact formula and is typically calculated using approximations.

Formula for Ellipse Perimeter

While there's no simple exact formula, a commonly used approximation for the perimeter of an ellipse is:

\[P \approx \pi(a+b)(1 + \frac{3h}{10 + \sqrt{4-3h}})\]

Where:

• \(P\) is the perimeter of the ellipse
• \(a\) is the length of the semi-major axis
• \(b\) is the length of the semi-minor axis
• \(h = (\frac{a-b}{a+b})^2\)
• \(\pi\) (pi) is approximately 3.14159

This formula, known as Ramanujan's approximation, is accurate to within 0.04% for any ellipse.

Calculation Steps

1. Identify the lengths of the semi-major axis (\(a\)) and semi-minor axis (\(b\)) of the ellipse.
2. Calculate \(h = (\frac{a-b}{a+b})^2\).
3. Substitute these values into the formula: \(P \approx \pi(a+b)(1 + \frac{3h}{10 + \sqrt{4-3h}})\).
4. Compute the result to get the approximate perimeter of the ellipse.

Example

Let's calculate the perimeter of an ellipse with a semi-major axis of 5 units and a semi-minor axis of 3 units:

1. Given: \(a = 5\) units, \(b = 3\) units
2. Calculate \(h\):
   \(h = (\frac{5-3}{5+3})^2 = (\frac{1}{4})^2 = 0.0625\)
3. Apply the formula:
   \(P \approx \pi(5+3)(1 + \frac{3(0.0625)}{10 + \sqrt{4-3(0.0625)}})\)
4. Calculate:
   \(P \approx 25.5265\) units

Visual representation:

Therefore, the perimeter of the ellipse is approximately 25.53 units.