I know that this is an old topic and the previous answers have been educational and revealing to me; I'm fairly new to Illustrator. I discovered this thread in my search for the same thing the OP sought; a way to divide a circular path into precise arcs by superimposing another shape. I couldn't find an answer that satisfied my needs but, luckily, I eventually figured it out. Here's how:
1) Create a circle with a diameter equal to the radius of the desired arc. Give it a stroke but no fill. Turn on Smart Guides. This next part will be a bit tricky, if you haven't done this before, so read this next step before attempting it.
2) Select the Polar Grid tool from within the Line Tools palette. Position the cursor over the center of the circle (that's why you have Smart Guides on) and Alt-Shift-drag (or Option-Shift-drag) out from the center until the polar grid's outer circle is larger than your original circle. Holding down the Alt-Shift (or Option-Shift) keys will constrain the polar grid to the center of your original circle and keep it perfectly round. DO NOT let go of the mouse button. (In the image below, I have let go of the Option-Shift)
You'll note that the polar grid contains a number of concentric circles and a number of radial dividers. Keep your mouse button down with your mousing hand but let go of the Alt-Shift (or Option-Shift) keys with the other; the polar grid will move but do not let this throw you and do not stop pressing the mouse button. Now, using your non-mouse hand, press the down-arrow key until there are no concentric rings inside the polar grid's outer circle. Then use the same hand to press the right-arrow or left-arrow keys to increase or decrease the number of dividers to create the number of divisions you want. If you want a 22.5˚ arc, for example, divide 360˚ ÷ 22.5˚ to get 16, the number of radial dividing lines you'll need. You are still holding down the mouse button, right?
Once you've cleared the concentric circles away and arrived at the number of divisions you need, press the Alt-Shift (Option-Shift) keys again. When you do, the polar grid will now pop back to the center of the original circle and will be constrained to perfectly round. Now let go of the mouse button to create the polar grid. You should now have your original circle overlaid by a polar grid consisting of the number of dividers you need and one outer circle.
The polar grid is actually a group consisting of to groups; the outer circle and the group of radial lines that constitute our dividers. Using the Layers Panel, twirl open the polar grid group, select the outer circle and drag it to the trash bin in the Layers Panel to delete it. Leave the radial dividers grouped.
3) Now select both the original circle and the group of radial lines and use the Pathfinder to cut the original circle into segments. There are two ways you can do this:
A) Use the 'Outline' tool, which will cut all the lines and arcs
into discrete path segments OR
B) Use the 'Divide' tool to cut the original circle into pie-shaped objects (in this
case, you will need to select the point of the pie-shaped object with the Direct
Selection tool (white arrow) and delete it to create the arc-shaped path)
Whichever tool you use, you'll need to 'Ungroup' the resulting paths (or shapes, in the case of the 'Divide' tool) in order to select and move them and create the objects you desire. If you use the 'Outline' tool, it will remove the stroke, so you'll have to apply a new one in order to see the path segments. The 'Divide' tool will leave the stroke intact but, again, you'll need to select and delete the center point of each segment in order to reduce the shape to the path you desire.
I hope that this will prove useful to others. Best Wishes to All and, as Elrond would say "Nasië i eleni síluva tielyanna!" ("May the Stars Shine Upon Your Path!")
