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

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.

This was my pervious video featuring ellipsoids:

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).

Bonus content and a deleted scene are available on my Patreon.

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.

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

As always: thanks to Jane Street who support my channel. They're amazing.

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

[JacekJurewicz]

My lazy approximation would be 4a :) The more eccentric the ellipse, the more accurate it gets.

[MrKalerender]

"I know just enough mathematics to be dangerous" - I feel this should be a tshirt.

[misterguts]

15:36 3 Blue 1 Brown's pi is sort of like the Clippy of mathematics:
"It looks like you're trying to find the perimeter of an ellipse!"

[Alan_Stinchcombe]

Matt, engineers frequently use the "root mean square" to describe expressions like SQRT((a^2 + b^2)/2).

[mingxizhang]

15:30 Matt: *slaps Pi”

“This bad boy can fit an infinite series of fractions in it’

[random]

"Who's having an ellipse that is 75 times as long as it is wide?"
An Oort Cloud comet has entered the chat.

[ujustinree]

That moment of realization for 2*pi*r where he says "wait a minute!" is so well timed with the realization for the viewer.

[mixbaal]

I am almost 60 years old. I love mathematics and I never, never imagen if somebody could make me laugh watching a math video. Well you did. Mathematics are so amazing, fun and funny too. Thank you so much for this 20 mins. Cheers!

[dottormaelstrom]

If you actually want the answer to "why don't we have a formula", it is simply that the perimeter of an ellipse is the line integral of its parametrisation: an ellipse is the set {(a cos(t), b sin(t)): 0<t<2pi}, which means that its length is the integral from 0 to 2pi of its derivative's norm, ie ∫sqrt( a²sin²(t) + b²cos²(t))dt, which is not an elementary function (meaning it's not an easy thing in terms of log, trig, exp and all the "usual functions", it's called (coincidence?) an "elliptic integral") for general a and b; for example it is for a=b=r and it gives the answer 2 pi r.

[ayrtonsenna]

" if ramanujan made 1 major mistake with their mathematical career, it was having it in the past" -matt parker, everybody

[kruks]

There aren't enough comments about how wonderful that 3Blue1Brown π cameo was.

[Astromath]

13:06 Well, because an object in free fall isn't really tracing out a parabola but instead a highly eccentric elliptic orbit around the earth's gravitational centre, you might in fact need such high eccentricity

[sproga_]

Loved the little 3Blue1Brown reference.

[KrazyKyle-ijvb]

8:35 "His mistake was doing math in the past."

Honest mistake, we'll try to do better next time.

[vaibhavchandra]

6:45 thats called the 'root mean squared' value. Read the words in opposite order and you will know why. Very useful in kinetic theory of gases as well as calculations of alternating current.

[edoardoferretti]

The interesting fact I noticed about the "bouncing" approximation is that for certain values of ratio they give a 0% error

[vikraal]

Whenever Mathematicians are scratching their heads on a problem, a wild Ramanujan appears

[gengis]

I just realized that my math teachers frightened me in knowing formulas of perimeter, area and volume of nearly anything, omitting to tell that one was missing.

[Maseiken]

"So what are the traits of an ellipse?"
"Oh well there's the major and minor axes, two focal points, an eccentricity and h."
"What's h?"
*leaves*

[suomeaboo]

If I had a nickel for every time Matt Parker called an ellipse an "eclipse", I'd have two nickels. Which isn't a lot, but it's weird that it happened twice.