Polygons: why are three and four different from other "sidegons"

Solution 1:

According to the OED's citations, square was first used 1300-1400; triangle 1398; quadrangle 1398; rectangle 1560;

then hexagon 1560; polygon 1560; pentagon 1570; octagon 1594; trigon 1600 (basically all around the same time) It looks like we adopted a neat Latin naming convention as we found interest and practical use for more figures with an increasing numbers of sides:

-gon, comb. form

1652 News from Lowe Countreys 2 For 'tis not..Trigonall, or Pentagonall, Or any of the Gones at all OED

Etymology: < classical Latin -gōnon (in e.g. octagōnon octagon n.), -gōnum (in e.g. trigōnum trigon n.) ...

Compare French -gone.

It's unreasonable to think we'd rename our three- and four-sided shapes—triangles, quadrangles, squares, and rectangles. We just "gone" off after that: new shapes need new names.

The OED has this telling early citation for hexagon:

1570 H. Billingsley tr. Euclid Elements Geom. iv. f. 124 We may in a Hexagon geuen either describe or circumscribe a circle.

I think architectural and scientific interest in geometry (and Euclid in particular) played a role in adopting -gone.

An interesting early -angle "exception" is the five-pointed star, a pentangle (1390):

A pentagram; a talisman or magical symbol in the shape of or inscribed with a pentagram OED

In other words, a pentangle isn't a pentagon.