Mensa Ellipse Geometry Solution

 

The ellipse is first re-drawn as shown with the major axis = 2a.  Then draw the minor axis and let its length be 2b.  The area of the ellipse A (e)= p ab.  Note that the green region comprises two unabridged quadrants and two right triangles. 

The area of the two quadrants is:  ½ (p ab).   The area of the two right triangles = 2 (½ ab)  = ab.  

Then the rea A of the green region is: ½ (p ab) + ab.   Given there is only one Pythagorean triple (9, 12, 15) in which the largest number is 15 we must have: a =12, b = 9. 

 Then: A = ½ (p · 12 · 9) + 12 · 9 =   54 p   + 108

Or:   54 (p   + 2)