If an ellipse has semi-major axis length a, and a circle has radius a, and you walked along their boundary, which one would be longer? A circle's circumference is calculated using $2\pi r$, but I don't know the equivalent for an ellipse. Probably involves integration. Is there a general rule or will it vary from case to case?

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  • $\begingroup$ What is the semi-minor axis? You can find much information in the Wikipedia article on Ellipses: en.wikipedia.org/wiki/Ellipse. In particular, look under "circumference". Are you looking just to compare the circumference of a circle with that of an ellipse or to calculate exactly? $\endgroup$ – Micapps Jan 10 '16 at 10:33
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    $\begingroup$ Not to come off as rude, but have you tried Googling "perimeter of ellipse"? Haitorically, this was a very important question, and the solution was one of the major early successes of calculus. $\endgroup$ – David H Jan 10 '16 at 10:33
  • $\begingroup$ @Micapps In particular, I wonder whether a planet moving in an elliptical orbit would go further than one moving in a circular orbit. I'm not really looking to calculate exactly, just make a general comparison. $\endgroup$ – user13948 Jan 10 '16 at 10:42
  • $\begingroup$ @Karacoreable It still seems you're missing information. What is the semi-minor axis? If it is very small the perimiter will be less than the circle and if it is large it will be larger. $\endgroup$ – Micapps Jan 10 '16 at 10:44
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    $\begingroup$ @micapps: The semi-minor axis is, by definition, smaller than the semi-major axis. So the circumference of the ellipse will always be shorter than the circumference of the circle. $\endgroup$ – TonyK Jan 10 '16 at 10:54

If the semimajor axis of the ellipse is identical to the radius of the circle, then the perimeter of the ellipse is smaller. One way to see this is by continuously increasing the semiminor axis to match that radius as well. This will stretch your ellipse to a circle, and that stretching will only make things (lengths as well as areas) larger, never smaller.

Actually computing the perimeter of an ellipse involves elliptic integrals, which is the reason why there is no direct analogon to the simple $2\pi r$ you have for circles.

I wonder whether a planet moving in an elliptical orbit would go further than one moving in a circular orbit.

If that's what you are after, then likely the perimeter is of little interest. You should ask about this explicitely (but not in this question here, and only after checking existing answers). If you do so, make sure to be precise as to what you fix in your comparison. Is it the energy of the planet, or the perihelion distance, or the orbit period, or whatever.

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