Question

the perimeter of an ellipse with an area equal to 12pi square units if the non opposite vertices are 5units apart

Answer 1

http://www.mathwarehouse.com/ellipse/area-of-ellipse.php

Answer 2

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
http://www.mathsisfun.com/geometry/ellipse-perimeter.html

Here is an applet that let's you find all properties of an ellipse
http://www.cleavebooks.co.uk/scol/callipse.htm
enter the 2a= 9.6883
enter 2b = 2.47721..
and press calculate to get a perimeter of appr. 20.87

Answer 3

22.21