Carpentry and Woodworking - determining side length in polygons

I hope I can state my question correctly. I would like to know if there is a formula for determining how long the sides of a polygon should be based on the inside dimension. For instance, if I want to create an octagon that is 6" inside, how long do I have to cut the individual pieces that will make up the sides. I can't imagine that professional woodworkers would use a trial and error method to create polygons of any size or number of sides.

I don't remember the formula, but AutoCAD has a tool that calculates that for you.

If you want the sides of the octagon to touch the 6" circle, so the points are just outside the circle, the side lengths are 2.485". I think this is what you described.

If you want the octagon points to touch the circle, so all sides of the octagon are within the 6" circle, the sides should be 2.296" long.

In fractions, those are 2-1/2" and 2-5/16", respectively.

kpm, I am so glad you converted this. One of my helpers is a retired mechanical engineer, and we share cad drawings for diagnosis work (I am a nail driver). He started sending me drawings in 4 decimal points. Imagine using 10,000th's of an inch clearance in framing!! I set my side of the program for slightly less tolerances and in fractions, so I can understand what he is talking about.
As far as an octagon goes, I usually draw my circle, draw two intersecting lines at 90 degrees, and then divide each one and draw two more lines. Connect the dots and don't measure. Not as scientific as the pythagorean theorem, but it works.

Use the radius of the circle to get the sides of a hexagon.( six sides)

There's design and then there's implementation. There are handbooks that provide the formulas and PC programs that will draw it for you.
For polygons with at least an even number of sides (like 6 and 8) drawing the circle on the actual material and dividing it up is fairly easy. For an odd number of sides (like 5 and 7) you will likely need a protractor as a start.
Now when it comes to cutting, there's the challenge. Saw blades (for wood) and OA torches (for steel) have a kerf width and may deviate from a straight line slightly. Even the pencil or pen used to mark the cut has a width that increases as the marking material wears down. This is where cutting oversize and trimming back to the proper line occur.

I appreciate the replies. I have autocad but I am totally unable to use it. It is a bewildering array of buttons and tool bars. I managed to draw a circle but could not even figure out how to draw a polygon of any shape. I guess I could get an "autocad for dummies" book or something.

For those of you who like math...

Sides of a regular polygon: http://www.mathopenref.com/polygonsides.html

There are 2 different formulas, depending on whether you know the radius (of a circle that would surround the polygon) or the apothem (the distance from the center of the polygon to the center of one of the sides).

Those formulas worked like a champ! Thanks so much for the tip.

Math formulas never cease to amaze me!