First, reconfigure the artwork.
Use guides to determine the center of the circle and place one polygon at the vertical center.
This step is a bit difficult to show in screenshots. With the pen tool click just once at the circle's center to place an anchor point there. All you want is the anchor point, not a shape or path. With that anchor point still selected, shift-click to select the Polygon as well and group the polygon with that single anchor point.
Now choose Effect > Distort & Transform > Transform
First, of course, tick the preview option. Then click the bottom center point for the 9-point origin box enter a number greater the 0 in the Copies field, and then enter an angle value until you get the offset you are looking for. Click OK.
The polygon should still be selected, choose Object > Expand Appearance. This will create independent shapes for the effect you just applied. You can now use the Direct Selection Tool (white arrow) to click and delete any polygons you do not want. You can also delete all the duplicate single anchor points in the center of the circle.