Count 'em All 6!

We'll divide the grid into 3 parts, A, B and the shaded part: So the number of squares in the original grid is equal to the number of squares in grid A, plus the number of squares in grid B, minus the number of squares in the shaded grid.

Using the formula from this note , the number of squares in grid A ( $a=13$ , $b=21$ , $a$ is the width of the grid while $b$ is the height of the grid, but since $a<b$ we swap them around so it becomes $a=21$ , $b=13$ ) is $\frac{13\times14\times(3\times21-13+1)}{6}=1547$

Similarly the number of squares in grid B ( $a=25$ , $b=10$ ) is $\frac{10\times11\times(3\times25-10+1)}{6}=1210$

The number of squares in the shaded grid ( $a=13$ , $b=10$ ) is $\frac{10\times11\times(3\times13-10+1)}{6}=550$

Hence, the number of squares in the original grid is $1547+1210-550=2207$ .