Setting exact angles is possible by:
1. Select your object
2. Choose the rotation tool
3. Option/Alt Click on your object
A dialog box will appear, you can rotate an exact angle by entering the amount.
Think of a polygon like a circle with a limited number of sides. A circle has a circumference of 360 degrees. Takes the number of sides and divide it by 360 and you will have the angle required to place an object around the polygon.
But if I want to rotate around a pivot point with an exact angle (that is not 45° etc?)
Feels like this is a simple operation that shouldn't require creating an other object.
Rotate tool: Option click on the point you want to rotate the object around. This calls up the dialogue box and you can enter the angle.
Steve is correct. The rotate tools allows you move the center point otherwise called pivot point. When you hold down the alt or opt key and click the new location, the rotate dialog box will pop up, assuming it is active.