Octagons - 8 sided - will not sew together without blank spaces, so you need those triangles.
Hexagons - 6 sided - will sew together without blank spaces. You sew each hexagon in halves and put them up on a design wall, and then sew the rows together, with one hexagon with the top half and one hexagon with the bottom half and alternate across. Then you sew the rows together to put each hexagon with its mate.