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

From Standup Mathematician.

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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