I believe I gave you sure fire way of doing this and controlling it in the future for this very reason, but this is the method I would use to both crete the cogged wheel and to be able to change it at will without redrawing it or deleting any thing just change the settings in The Effects>Distort and Transform>Transform you can change it to how many cogs you want just changge the umber of copies  and the change the other settings to accommodate the change.

I don't know if there's a formula for producing regular polygons of the same side length and with the number of angles as a variable.

If so it would certainly simplify the drawing of gears.

Anyone? Jacob maybe?

I don't know if there's a formula for producing regular polygons of the same side length and with the number of angles as a variable.

A few years ago, I wrote a simple Javascript for exactly that. Quickly defines a regular polygon by desired side length.

But one can do much the same thing a little less quickly (Illustrator's rounding errors notwithstanding) by:

(SmartGuides on)

1. Polygon Tool: Click. In the dialog, specify the number of sides.

2. WhitePointer: Select one segment. Copy. PasteInFront.

3. Document Info palette: Note the length of the pasted segment.

4. DoubleClick the Scale Tool: Key the desired value into the Uniform field, followed by two zeros and a division mark, followed by the value noted in the Document Info palette. Click OK.

5. Select the whole polygon. Transform Again. (Ctrl D).

The trick would then be to replace each segment with the desired path. Several scripting approaches could be used for that. (I'd likely use a modification of my ReplaceWithSymbol script--complicated by Illustrator's insistence upon dimensioning Symbols by their preview bounds.)

But you can accomplish all this with Transform Effect:

1. Draw the tooth in a horizontal orientation, with the tooth pointing downward.

2. Control Panel: Select and copy the value in the Width field.

3. Effect>Distort & Transform>Transform.

Set the 9-point transformation center icon to lower left.

Paste the value into the horizontal Move field. (There should be a Percentage Of Original toggle for these fields, just as there should be Absolute size fields in Transform Each.)

Copies: Enter one less than the number of teeth desired.

Rotate: Enter "360/", followed by the number of teeth desired.

But the reason I have not responded to this thread until now is that the original question is too poorly stated. If this is for illustration purposes only, a Pattern Brush (using Approximate Path for the Fit option) should do fine. But Pattern Brushes do "wedge" distort their side tiles, so if this is actually needed for geometric accuracy, the Pattern Brush would not serve. And if dead-accuracy is, in fact, needed (as if for an actual working drawing for manufacturing), other considerations may come into play. For example, the actual shape of sprocket teeth may be required to change dependent upon how many teeth are on the sprocket. (The countershaft sprocket and rear-wheel sprocket of a motorcycle fit the same chain; but the shapes of their teeth differ slightly because the chain runs in a much tighter bend around the countershaft sprocket.)

JET