These shapes have four sides and 360° of interior angle goodness.

Suppose \(ABCD\) is an isosceles trapezoid with bases \(AB\) and \(CD\) and sides \(AD\) and \(BC\) such that \(|CD| \gt |AB|.\) Also suppose that \(|CD| = |AC|\) and that the altitude of the trapezoid is equal to \(|AB|.\)

If \(\dfrac{|AB|}{|CD|} = \dfrac{a}{b},\) where \(a\) and \(b\) are positive coprime integers, then find \(\large a^{b}.\)

The numbers \(3,4,\) and \(6\) denote the area enclosed by their respective triangles.

What is the area of the yellow region?

What is the minimum internal surface area I can achieve?

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