Arc Length in Ellipse

can anyone tell me what is the length of circumference of an ellipse? actually from symmetrical point of view between a circle and ellipse i guessed it to be pi(a+b). i tried to use arc length formula but stuck in a lengthy integral. so i need to get the correct answer in the proper way.

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Easy Math Editor

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Comments

There is no easy answer. The circumference has to be expressed in terms of the complete elliptic integral of the second kind $E(e) \; = \; \int_0^{\frac12\pi} \sqrt{1 - e^2\sin^2\theta}\,d\theta \qquad 0 < e < 1$ and the circumference of an ellipse is $4aE(e)$ , where $a$ is the semimajor axis and $e$ the eccentricity. A good approximation is $\pi(a+b)\left(1 + \frac{3h}{10 + \sqrt{4-3h}}\right) \qquad h = \tfrac{(a-b)^2}{(a+b)^2}$ so your guess was a good approximation for not too eccentric ellipses.

Mark Hennings - 7 years, 7 months ago

thanks for letting me know

Shubhabrota Chakraborty - 7 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$