I’ve been experimenting with methods to calculate the perimeter of an ellipse, and I came up with a formula that approximates it using a summation of small line segments, and then defines the exact perimeter as the limit as step size h approaches 0.
Approximate Formula
Let the ellipse be defined by: $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$
The top half of the ellipse is described by: $$ y(x) = b \sqrt{1 - \frac{x^2}{a^2}} $$
Let h be the step size, and define sample points: $$ x_i = -a + i \cdot h \quad \text{for } i = 0, 1, 2, …, N-1 $$ where N is the number of steps before overshooting a.
Then the approximate perimeter is: $$ P(a,b,h) \approx 2 \left( \sum_{i=0}^{N-2} \sqrt{h^2 + \left(y(x_{i+1}) - y(x_i)\right)^2} + \sqrt{(-x_0 - x_{N-1})^2 + \left(y(-x_0) - y(x_{N-1})\right)^2} \right) $$
Limit Definition
Taking the limit as h approaches 0, the exact perimeter is defined as: $$ P(a,b) = \lim_{h \to 0} P(a,b,h) $$
Question
This formula seems to converge to the correct perimeter very well — even outperforming Ramanujan’s approximation in many cases. But I want to know:
• Is this a mathematically valid way to define the exact perimeter of an ellipse?
• Does this formulation already exist elsewhere under a different name?
• If not, does this method qualify as a rigorous perimeter formula by limit definition?
Here is a Google Doc with full explanation, proof, and comparison data that outlines the discovery in more detail.
