Making circular shapes and patterns in Illustrator is straightforward using a simple technique. Here's how to save yourself some time.
A simple start
Let's introduce the technique first. In this example we want to create a circular pattern of circles. First draw a circle on the canvas using the Elipse Tool (shortcut L). My circle has no fill and a stroke of 1pt. Now with the circle selected select the Rotate Tool (shortcut R). Click where you want the centre of the rotation to be. You should see a small turquoise target icon appear. Now rotate the shape by clicking above the shape and dragging. If you hold ALT the shape will duplicate.
Now we use transform again to replicate the shape over and over again to get our circular pattern. You can Transform again by hitting Apple + D on a Mac or CTRL + D on windows. If you struggle with the Shortcut go to Object > Transform > Transform Again. Watch the video below to see this process in action.
Using the technique
You can apply this technique to any shape or object that you draw on the canvas. Holding shift as you rotate your object will confine it to 45 so you can be sure you get a perfect circle.
I've used this technique here to quickly create a wallpaper/textile design. Thanks to the technique it took about five minutes.