# A paradox in Geometry concerning the perimeter of an ellipse

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#### waveplumber

##### Junior Member level 3

Can someone compute the exact value of the perimeter of an ellipse of semi major axis a = 8.0 m and semi minor axis b = 7.5 m and tell me the results? From my calculations, the answer is less than the perimeter of a circle of radius 7.5 m, which I think does NOT make sense because the later lies inside the boundaries of the ellipse in question.

Thanks

#### Kral

waveplumber,
From "Handbook of Mathematical TYables and Formulas" by Burrington:
p = 2Pi (Sqrt((a^2 +b^2)/2)) (Approximately)
.
p = 4aE (Exactly)
where
E = Complete elliptic integral of the first kind, using
. k = Sqrt(a^2-b^2)/a
. Phi = Arcsin(k)
Regards,
Kral

#### waveplumber

##### Junior Member level 3

Thanks Kral. I had used the second formula [p = 4aE(e)] but my worry is that the perimeter of the ellipse (a =8.0, b =7.5) comes out to be less than that of a circle of radius 7.5. This is counter intuitive so I'd like to verify my calculations or have an explanation for the apparently paradoxical situation. Can you look into that?
Thanks

#### flatulent

You can ease your emotions by drawing the ellipse and then us a string to go around the plotted line and then measure the length of the string. Empirical methods sometimes are useful.

#### waveplumber

##### Junior Member level 3

No, no no, patulent. I prefer to do this by analytical methods. Thanks though.

#### Kral

waveplumber,
here are my calculations:
k = SQRTSQRT(1-(b^2)/(a^2)) = 0.347985
Phi = ArcSin(k) = 20.36413 degrees
E = 1.522089 (Interpolated value from table)
p = 4*a*E = 48.70683
.
p of circle radius 7.5 = 2*Pi*7.5 = 47.12389
p of circle radius 8.0 = 2*Pi*8 = 50.26548
.
So the caluclate perimeter falls in between the two circle perimeters.
Regards,
Kral

#### btwang

##### Newbie level 5

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#### waveplumber

##### Junior Member level 3

Thanks Kral. I was using Matlab to compute E(e) and I just realized that the parameter that Matlab requires is the modulus of the elipticity e

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