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I recently came up with a formula for building octagons in minecraft, you take the length of the diagonal wall and you get the length of the straight wall, ie. 55-diagonal, 77-straight. The ones that are perfect are; 5x7, 10x14, 15x24, 20x28, 25x35, 30x42, 35x49. you can figure out the rest yourself with a calculator. You can use this formula for building epic octogons that look cool when used right.
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But there are better solutions.
These all work pretty good...
flat diagonal diameter(final)
7 5 18
10 7 25
14 10 35
17 12 42
If you want to find large ones run the formula 'd=12;sqrt(d^2*2)' where 'd' is the diagonal integer. If the number comes out close to an integer you have another good ratio of values. There aren't a lot of them! The ratio is by the way not 1.4 It is 1.41421... or the square root of 2
But remember... These are the mathematical lengths. they represent the distance from one vertex to another. Which if you have one block vertices means it represents the lengths from the center of that vertex block, to the next.
As such if you are building the flat octagon face a length of 10 will produce a line of blocks that is 11 long! Similarly the diagonal for that octagon will be 8 blocks diagonally, not 7! But the first and last block of each edge will be shared with the next edge, going around the octagon.
The diameter of the octagon will finish
face + diagonal * 2 +1
Which for the 10 flat become 25 blocks total!
Actually for my project I was building a 3-dimensional rhombicuboctahedron.
another useful thing to know is that the TOTAL WIDTH from side to side can be given by:
(n/sqrt(2)*2)+n+2
depending on what size area you want to fill, or how long you want the sides, you can get any shape that is pretty much accurate..
(2xdiagonal length + Normal side Length) + the thickness of the material, twice (assuming you're measuring outside dimensions)