Why is there no equation for the perimeter of an ellipse‽

Video Description

Jane Street are still offering amazing internships: https://jane-st.co/SUM-internships A Podcast of Unnecessary Detail is out now: https://festivalofthespokennerd.com/podcast/ Come for the Unnecessary Detail, stay for the A Podcast Of. These are my approximation equations: perimeter ≈ π[53a/3 + 717b/35 - √(269a^2 + 667ab + 371b^2)] perimeter ≈ π(6a/5 + 3b/4) If you can do better, submit it to Matt Parker's Maths Puzzles. https://www.youtube.com/watch?v=qJSHdJX6gB0 https://www.think-maths.co.uk/ellipsepuzzle This was my pervious video featuring ellipsoids: https://youtu.be/l51LcwHOW7s You can buy the ellipse from this video on eBay. I've written on my two new equations and signed it. All money goes to charity (the fantastic Water Aid). https://www.ebay.co.uk/itm/363096345270 Bonus content and a deleted scene are available on my Patreon. https://www.patreon.com/posts/41274351 Huge thanks to all who sent in a recording of them singing "A total ellipse of the chart." Sorry I could not include everyone. These are the people in the video: Helen Arney Steve Hardwick Victoria Saigle Andrew McLaren Fractal Macey Sören Kowalick It all started because of a request I put out on twitter. https://twitter.com/standupmaths/status/1301252952930299904 CORRECTIONS: - So far the only times (I've noticed that) I say "eclipse" instead of "ellipse" are 5:01 and 05:26 which was just after talking about my wife who is a solar physicist. So I think we split the blame 50/50. - It seems everyone but me recognised the Root Mean Square. I think I only associate that with current for some reason! Thanks all. - Let me know if you spot any other mistakes! Thanks to my Patreons who meant I could spend about a week trying to find approximations for the length of ellipses. "Are you still working on that?" Lucie would - rightfully - ask over the weekend. "I'm going the extra mile for my patreon people!" I would reply. Here is a random subset of those fine folks: Benjamin Richter Louie Ruck Matthew Holland Morgan Butt Rathe Hollingum Jeremy Buchanan Sjoerd Wennekes Barry Pitcairn James Dexter Adrian Cowan https://www.patreon.com/standupmaths As always: thanks to Jane Street who support my channel. They're amazing. https://www.janestreet.com/ Filming and editing by Matt Parker Additional camera work by Lucie Green VFX by Industrial Matt and Parker Music by Howard Carter Design by Simon Wright and Adam Robinson MATT PARKER: Stand-up Mathematician Website: http://standupmaths.com/ US book: https://www.penguinrandomhouse.com/books/610964/humble-pi-by-matt-parker/ UK book: https://mathsgear.co.uk/collections/books/products/humble-pi-signed-paperback