Squares fitted in regular polygons

The process I initially follow , is to start with the regular polygon that I want to insert the squares in, draw a loose point in one of the sides and repeat on the other sides (same length). I draw the squares and move the original point until I find the meeting point of the squares. I use GeoGebra to make the drawings.

The triangle, 3-gon

I can't fit 3 squares in a triangle following this process, but still quite interesting geometrically.

The square,4-gon

No gaps left in a square.

Hexagon,6-gon

In a hexagon the squares meet in the midpoint of the square.

The hexagon and golden ratio using the previous template.

Heptagon,7-gon

Octagon, 8-gon

Square and square root 2.