# Does an ellipse or circle have greater circumference?

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?

• 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? – Micapps Jan 10 '16 at 10:33
• 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. – David H Jan 10 '16 at 10:33
• @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. – user13948 Jan 10 '16 at 10:42
• @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. – Micapps Jan 10 '16 at 10:44
• @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. – TonyK Jan 10 '16 at 10:54

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.