You get a circle with radius r. Its area is πr². You get an ellipse with major and minor axes a and b. Its area is πab.
The circle’s circumference is 2πr. The ellipse’s circumference is π(a+b), right? No.
I’m incredibly late to the party, but I just learned that there is no closed formula for the circumference of an ellipse. This blew my mind. I really felt like one of “the lucky 10000“. I also felt like sharing it with you, in case you cared to join me.
This launched me in an exploration which culminated in this paper, that I’m reading avidly. There are so few of these “scientific papers” that are written with the reader in mind. I can only imagine what a world of academics who really cared about being understood — rather than being published — would look like.