# Ellipse semi-major axis

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I know that the area of an ellipse is 1403761773.43497 and its circumference is 145485.418131498.

Is there any way to calculate from those numbers that the semi-major axis is 26534.9039306654?

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There is no straightforward method for calculating the circumference of an ellipse from the two semiaxes, so there is also no straightforward method for determining the length of any semi-axis from the area and the circumference.

Given the area and the suspected size of the major semiaxis, the small semiaxis would be 16839.3769767847.

Wikipedia suggests that at this proportion of the major to the minor semiaxis the circumvmference U = pi*(a+b) yields an error of 1%. So for the current assumed configuration, U should be 136264.322253585, which is more than 1% outside the originally stated circumference

For clarity: close, but no cigar

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Supposing that I correctly calculate the integral to obtain the circumference and the area (I'm not sure right now), is there any numerical method that I can try?

EDIT: I used the formula

C= pi * (a + b) [ 1 + sum...

to calculate the circumference and my above result (145485.418131498) is good, while the area is wrong.

Edited by Cristiano
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15 minutes ago, Cristiano said:

Supposing that I correctly calculate the integral to obtain the circumference and the area (I'm not sure right now), is there any numerical method that I can try?

That depends on how precise you need the results to be. As I've said, up to a ratio of 3:2 for the major to the minor semiaxis, the error is smaller than 1% for U = pi*(a+b). Since pi appears in both formulae for circumference and area, then A/U = ab/(a+b). This is the most I can offer right now...

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I meant a recursive formula like: a = A/U * (a + b) (I wrote just an example for clarification).

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