determining the length and cut angle of a 20x20 octogan?

Question

Let's say you have a 20x20 foot area that you want to construct an octagonal privacy fence using 4x4 posts. How do I determine the length of each panel and at what angle do I cut the ends? Is there a genreal mathmatical formula to determine the lengths given any area?

Answer 1

So... You have a 20x20 foot area, and you want to fit a regular octagon inside of that space, making the octagon as large as possible. You want to know how long each side of the octagon should be. Is that what you are asking?

In other words... You want to fence in a 20x20 foot area, leaving the corners of the area outside the fence. You also want all 8 sides of the fenced area to be equal.

The quick answer is:
Each side is 8.284 feet long. (that's 8 feet, 3 7/16 inches)

There is a fixed number that gives the ratio between a side of the octagon and the side of the square area that you started with. That number is:
1 / (1 + sqrt(2)), which equals 0.414214

20 feet times 0.414214 = 8.284 feet

The four triangular spaces outside the fence will have sides with lengths of 5.858 feet, 5.858 feet, and 8.284 feet.

82.8 % of the total area is inside the octagonal fence, and 17.2 % of the area is in the triangular spaces outside.

Answer 2

The angles are straightforward: since you want the angle between the current board and the next to be 45°, you need to cut the corners so they're 45°/2 = 22.5° less than 90°, straight across the board. Try drawing and cutting out pieces of paper first, to get a feel for it before you start cutting wood.

I can help you figure out the area. Imagine you have an octagon drawn on graph paper so the figure is 3 lines high. Approximately, you will have the three horizontal and three vertical centre squares covered (making five in total), and the remaining four squares will be about half covered.

So if S is the length of a side, then A = S*S is the area of one of those small squares, and the whole figure would be
A = (S*S)(5+(4/2)) = 7S^2
That's **approximate**, but it will give you an idea. A deck made in an octagon with 8' sides will be about
7*8^2 = 7*64 = 430 square feet, more or less.

Answer 3

Divide 360 degree (all the way around) by the number of sides. How you cut the ends depends on exactly how you put it together. One joist between each panel with a V end or each panel having its own edge, etc.