Which region has a greater area? yellow region or blue region?

Consider my diagram. Let the side length be $1$ . By pythagorean theorem, we have,

$1=x^2+x^2=2x^2$ $\color{#D61F06}\implies$ $x=\sqrt{\dfrac{1}{2}}$

It follows that, $y=1+2\sqrt{\dfrac{1}{2}}$ .

The area of the yellow region is the sum of the areas of two trapezoids. We have

$A_{yellow}=2\left(\dfrac{1}{2}\right)\left(1+2\sqrt{\dfrac{1}{2}}\right)\left(\sqrt{\dfrac{1}{2}}\right)=2\sqrt{\dfrac{1}{2}}+1$

The area of the blue region is

$A_{blue}=\left(1+\sqrt{\dfrac{1}{2}}\right)(1)=1+2\sqrt{\dfrac{1}{2}}$

Therefore, the areas are equal.