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Making a Cog in Adobe Illustrator

First we need to create a circle, select the 'ellipse tool' ellipse tool from the main tools menu, click on the document and as you drag the mouse and hold down the shift key, constraining the circle.
 
Next we need to draw a rectangle and modify it using the 'direct selection tool' selection tool, to do this click on a paths corner point, once selected the selected point will appear a solid colour while all the other points will be white. Then drag the point using the mouse, once again if you hold the shift key while dragging, it will constrain the path points movement. Now select and drag the point from the other side until we have a shape something like this.
dragged points

Now we need to position the shape to roughly where we want it over the top of the circle. A handy tip is to drag two illustrator guides (one horizontal and one vertical) onto the document unlock them and select the guides and the circle and using the Illustrator Align Tool align vertical center and horizontal center.

rotate shape
Now select the modified rectangle and using the 'rotate tool' rotate tool click in the center of the circle while pressing down the alt key. This will open a panel where you can add the angle of rotation (fig1) then click the copy button as seen below.
rotation panel
After clicking the copy button press apple + D on a mac or Ctrl + D on a PC, this will duplicate and continue to rotate the shape, you should now have something close to this.
cog
Next select all the elements and in the pathfinder panel click on 'add to shape area' add to area this will combine all the separate elements into one.
 
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