Could someone help me work out the geometry for making a scaled wooden windmill?

I don’t know what values you have chosen to use to specify your windmill. I am going to assume that you have chosen some base length b for the side of the base octahedron and a for the length of the side of the upper octahedron. I am also going to assume you have specified the slant height h1 joining a and b.

It is convenient to work with the triangle formed by extending the two sides joining the two bases. If we call call h the height of the triangle, then using similar triangles, we get (h – h1)/a = h/b. Solve for h. You can now calculate the base angles using (b/2)/h = tan(base angle).

You can also compute the tilt angle of the walls. First calculate the distance g from a side b to the center of the octahedron, given by (b/2)/g = tan(22.5). Then g/h = cos(tilt angle).

It’s easier than it sounds. Cut the number of boards you want for the walls from the same exact template, tapered at the top. They’ll lean in just as you want. You could try it first with cardboard or construction paper if you want to experiment. I just did it with some scrap paper. It worked. That gives you the tower.

Don’t forget to taper the side cuts, angle will depend on how many sides you decide to use. 22.5 degrees for 8 sides, 15 degrees for 6.

It really doesn’t matter, as long as all the sides are straight and exactly the same. I would start therefore with a sketch of what you want it to look like, and cut the sides to suit. The finished article will be SLIGHTLY shorter than you drew, but it really won’t be noticeable.

Good luck with your project