pleas-proof-the-elipse-environment-formullah-p-pi-2a-2-2b-2-

Question Number 179593 by mathlove last updated on 30/Oct/22

p l e a s p r o o f t h e e l i p s e e n v i r o n m e n t

f o r m u l l a h

p = π \sqrt{2 a^{2} + 2 b^{2}}

Commented by mr W last updated on 31/Oct/22

w h a t d o y o u m e a n w i t h “ e l i p s e

{e n v i r o n m e n t}^{″} ? d o y o u m e a n t h e

p e r i m e t e r o f e l i p s e ?

Commented by mathlove last updated on 31/Oct/22

y e s

Commented by mathlove last updated on 31/Oct/22

e l l i p s e

Commented by mr W last updated on 31/Oct/22

t h e n y o u {c a n}^{'} t p r o v e, b e c a u s e {i t}^{'} s n o t

t r u e . i n f a c t t h e r e i s n o c l o s e d f o r m u l a

t o c a l c u l a t e t h e p e r i m e t e r o f a n e l i p s e .

Commented by mathlove last updated on 31/Oct/22

t h e n w h a t i s t h e e l l i p s e n v i r o n m e n t f o r m u l l a h

Commented by mr W last updated on 31/Oct/22

y o u {d o n}^{'} t u n d e r s t a n d m e ? i h a v e s a i d,

t h e r e i s n o f o r m u l a f o r c a l c u l a t i n g

t h e p e r i m e t e r o f a n e l i p s e! i . e . y o u

{c a n}^{'} t c a l c u l a t e t h e p e r i m e t e r o f a n

e l i p s e e x a c t l y t h r o u g h a f o r m u l a!

Commented by mathlove last updated on 31/Oct/22

g o o d t h a n k s

Commented by MJS_new last updated on 01/Nov/22

for the perimeter of an ellipse we have

p = 4 \underset{0}{\int^{π / 2}} \sqrt{a^{2} \cos^{2} t + b^{2} \sin^{2} t} d t

which has no exact solution

Ramanujan gave this very good approximation

p \approx π (a + b) (1 + \frac{3 λ^{2}}{10 + \sqrt{4 - 3 λ^{2}}}) with λ = \frac{a - b}{a + b}

both “ {easy}^{″} formulas

p \approx π (a + b)

p \approx π \sqrt{2 (a^{2} + b^{2})}

are not accurate

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