Try this method if it can satisfy your requirements :-)
Using Eclipse tool you can create perfect circle shape by pressing Command+Shift while dragging the mouse for its width and height or you can input in transform window in exact diameter you want.
Copy the first circle path then Paste it in back (Ctrl+C then Ctrl+B)
Note: I use yellow (first) and cyan(second) to differentiate each other. It is useful to see where is you path in Layer Panel which is on top of each other.
Use the scale tool to make the cyan path bigger. Press the Alt key then point to the right point of the circle. This is to control where the scaled circle which is the cyan starts from. Normally the anchor point starts from the center so we need to change it to achieve the desired shape. You notice the anchorpoint changes and a pop up menu will appear.
Change the value to the desired thickness of the crescent you want to do.
Now with both cyan and yellow selected. Open the Pathfinder window. You can see there MinusFront (refer to the illustration). Pathfinder > Shapemodes > MinusFront
last for the arcs (i mis-interpreted it that you like the moon)
you have many techniques to use but it boils down to Scissor Tool or Cutter.
Here are samples how to cut. Option A is cut with line and divide (pathfinder window). The step will be 2 circle paths + Draw 1 line to the right then click divide. delete the excess / residual path then Scissor Tool.
Option B Direct Scissor tool then delete un-wanted areas.