I think that there are 2 solutions, one for squares with a 45 degree tilt only and the other is for a square with any tilt.
45 degree tilt solution
First I tried a tilted square with 3 dots along each side (the sides of the square bisect dots on the grid). I counted the whole squares inside and there were 4. I then counted the triangles remaining and these totaled 4 whole squares. Therefore, the area of this tilted square must be 8cm squared. For this square the formula was d ( number of dots along 1 side) + 4
This also worked for a square with 4 dots along each side. There were 12 whole squares inside and 6 more could be made from the remaining triangles. Therefore, this time it was the whole squares + 6. For this square the formula was d ( number of dots along 1 side)+ 6
w = whole squares
3d= w+4
4d= w+6
(d-2)+ d=h where h = half squares
For squares with any tilt
Draw a larger square around the tilted so that the vertices of the tilted square touch the edges of the larger square, therefore creating 4 right angled triangles. The larger square is drawn straight i.e. using vertical and horizontal lines as sides.
Find the area of the right angled triangles, multiply it by 4 then deduct it from the area of the larger square. Your answer will be the area of the tilted square.