If I have a grid made of equilateral triangles, I can easily form larger convex polygons as a set of tiles in that grid. I believe this holds for some (but not all) tilings of non-equilateral triangles.
The same for quadrilaterals - it's obvious for squares, rectangles and parallelograms, but I believe it also holds for some other (but not all) tilings of quadrilaterals.
I'm pretty sure tiling a flat area purely with convex pentagons is impossible, so skip that case.
With hexagons, a tiled grid is possible, but the only way to make a convex polygon from those convex hexagonal tiles seems to be to use only one tile - no multi-tile convex polygons seem possible.
My speculation is that it's only possible to form a multi-tile convex polygon from convex polygon tiles if all those tiles have interior angles at all vertices of 90 degrees or less, at least for those vertices that are at a vertex or edge of that larger convex polygon.
Is that speculation correct? Is there a proof or disproof?