What a nasty question.
the area of the ellipse is abπ , where a and b are each 1/2 of the axes
so abπ = 12π --->ab = 6 or b = 6/a
let A(a,0) and B(0,b) be the two adjacent vertices
given AB = 5
√(a^2 + b^2) = 5
a^2 + b^2 = 25
a^2 + 36/a^2 = 25
a^4 + 36 = 25a^2
a^4 - 25a^2 + 36 = 0
a^2 = (25 ± √481)/2 = appr 23.4658 or appr 1.53414
a = 4.844157.. or a = 1.2386... ignoring the 2 negative values
then b = 6/a
if a = 4.844148, then b = 1.2386
if a = 1.2386 , then a = 4.844157
ahhh, summetrical answers
So we basically have the same ellipse, one horizontal, one vertical
The perimeter of an ellipse is one of the most challenging questions dealing with the ellipse.
It requires Calculus and finding the length of a curve.
Here is a link that has some approximation methods
Here is an applet that let's you find all properties of an ellipse
enter the 2a= 9.6883
enter 2b = 2.47721..
and press calculate to get a perimeter of appr. 20.87
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