The Perimeter and Area of an Ellipse
Finding the area and the perimeter of an ellipse might seem like similar problems at first glance, but in reality the two differ enormously in difficulty.
The area enclosed by an ellipse is $S=\pi ab$, where a and b are the semi-major and semi-minor axes. This can be derived directly from the standard equation of the ellipse.
No general closed-form formula for the perimeter of an ellipse has ever been found. To compute it exactly, one has to fall back on the following infinite series:
$$C=2\pi a [1 - (1/2)^2 (\frac{c}{a})^2 - ({1\cdot 3}/{2\cdot 4})^2{c^4}/{3a^4} - ({1\cdot 3\cdot 5}/{2\cdot 4\cdot 6})^2{c^6}/{5a^6}-...]$$
which can also be written as:
$$C = 2\pi a \sum_{n=0}^{\infty} { - [\prod_{m=1}^n ({2m-1}/{2m})]^2 {c^{2n}}/{a^{2n}(2n - 1)}}$$
The distance c is called the linear eccentricity of the ellipse, equal to the distance from the center to either focus.
Of course, if you don't follow all this, there's no need to be too discouraged, because the mathematician Ramanujan gave a fairly simple yet remarkably accurate approximate formula:
$$C \approx \pi [3(a+b) - \sqrt{(3a+b)(a+3b)}]$$
There is also another highly accurate approximation (reportedly good enough even for computing planetary orbits):
$$C=\pi (a+b)[1+\frac{3\cdot ({a-b}/{a+b})^2}{10+\sqrt{4-3({a-b}/{a+b})^2}}]\cdot [1+({22}/{7\pi} -1)({a-b}/a)^33.697]$$
Below are some reference values for the perimeter of an ellipse:
a——b————ellipse value
100—000——400.00000000
100—001——400.10983297
100—010——406.39741801
100—025——428.92108875
100—050——484.42241100
100—075——552.58730400
100—090——597.31604325
100—099——625.18088479
100—100——628.31853070
Translated automatically with claude-sonnet-5; all equations are reproduced verbatim from the source. Copyright remains with the original author.