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Sorry if the terms I used are incorrect, but I was wondering how I can divide a circle into several triangles, one for each side of the circle, as well as subdivinding those into quadgons like so:
I made the example on the left by hand (selecting each vertex and adding a face) which took way too long, and would clearly take too much time to do with other more complex circles.
I tried using subdivide and loop cut and slide, to no avail. So how can I do this quickly?
There are several approaches.
If you have a circle that is a triangle fan and you want to give it rings, just Circle Select C the edges that make up the "spokes" of the circle, Esc out of Circle Selection, and Subdivide W > "Subdivide".
In the Operator Panel on the left you can adjust the number of subdivisions.
If you don't yet have a triangle fan and need to create one, select the edge loop of your circle, Extrude E, then Scale S inward. Type "0" to scale in to the absolute center, Enter to commit. Remove doubles W > "Remove Doubles".
Note that the face normals may not be pointing in the direction you want. In the GIF above this is evident from the dark shading. You can remedy this by selecting the faces of your circle and pressing W > "Flip Normals".
When you create your circle there is an option for Fill Type. Set it to Triangle Fan and you will already have a circle made up of tris.
A less common approach, but good to know about.
If you have a circle that is an n-gon, with the face selected, press I and you can inset it inward. Finally, Scale S in to "0" and Remove Doubles W > "Remove Doubles".
If you're starting from a triangle fan, one more approach you can take is to bevel the center vertex to create the edge ring necessary to perform a Loop Cut. If you type in a near-zero value the 8 resulting center verts will be right on top of each other and you can easily patch the center hole later using Remove Doubles.
Using a Spin Tool.
Add a Circle with a triangle fan and subdivide (W-->Subdivide) one of its edge as many times as you like. Select the non-selected vertices (Ctrl+I) and delete them (X). Go to the Tool Shelf (T) on the left and press the Spin button located in the Tool bookmark. Then adjust the spin settings and remove doubles (with the whole mesh selected press W-->Remove Doubles).
Adding the loop cuts.
Add a Circle with no fill. Select all the vertices (A),extude them (press E), scale (press S) and input a very low value (like 0.0001) from your numerical keyboard. Now press Ctrl+R to add some loopcuts. Use a mouse wheel to set the number of the cuts. Deselect everything (A), then select the vertices in the middle (use a border select B) and merge them at cursor (W-->Merge-->At Cursor).
