Both long diagonals are drawn across a 3 by 3 grid of squares. Find the number of triangles.

Right triangles with their long side on one diagonal: 6 small, 4 over two squares, 2 over the whole grid, 12 per diagonal, 24. Triangles with their tip at the centre and their base on a grid line or edge: 4 on the inner lines and 4 on the outer edges, 8. Total $24+8=32$.