Which angle does my saw want?
The saw setting is 180° divided by the number of sides. Most miter saws read 0° for a square cut, so for an octagon you swing the blade to 22.5°. Measured from the board's edge instead, the same cut is 90° − 22.5° = 67.5°. That's the number a protractor, a table saw's miter gauge marked 90° for square, or a lot of web calculators give you. Both are right; they're measured from different lines. Check which way your saw's scale reads before you cut.
How to use it
Measure across the flats, not the corners. The width is from the middle of one side to the middle of the side opposite. With an odd number of sides there's a corner opposite instead, so use twice the distance from the centre to a side. If you know the inside size, say for a planter liner or a box that has to hold something, switch to Inside; the calculator adds a board width on each side to get the outside.
Board width is the face of each piece seen from above: the width of a frame's moulding, or the thickness of a box's sides.
Cut a test pair from scrap first. An octagon has 16 cuts, so a saw that's off by half a degree leaves the last joint 8° open. Cut two pieces, hold them together, and check the corner with a square or the angle you're after before you cut the good wood. Then set a stop block to the long point length so every piece comes out the same.
The total length of stock is every piece end to end at its long point. It doesn't include the kerf or the waste between cuts, so buy a little more.
A worked example
A hexagon planter 12" across the outside flats, from boards 3" wide: set the saw to 30° and cut six pieces 6 15/16" (6.928") long point to long point. Each one is about 3 15/32" (3.464") on the inside face, and you need about 41 9/16" of stock before kerf.
Squares, rectangles and splayed sides
A square frame is 4 sides at 45°. A rectangular picture frame is 45° too; each side is just its own length, long point to long point, so you don't need a calculator for the angle.
This calculator is for sides that stand straight up or lie flat. Sides that lean out, like a splayed planter or a flared tray, need a compound miter (a bevel as well as a miter), which it doesn't cover.
Turning a segmented bowl instead?
A segmented ring is the same polygon, cut a little oversize so it turns round. The Segmented Ring Calculator adds that allowance and gives you the strip to rip.