I am looking to put a roof over a octagon shaped patio. I have 4x6 posts at each corner. What size beams would I need to put a pergola styled roof above the patio to span the 12' distance in between the posts?
There are two considerations: 1) Structural, and 2) Aesthetics.
1) The pergola is small enough that you could span it with a 2x6. However, it will look weeny.
2) I’d use 4x6 (minimum) beams to match the posts. If you bolt them together they will want to “splay” outward. So, I’d use a gusset plate at the peak to help it keep its shape. Remember, steel connectors are basically “hinges”, so provide plate supports for rigidity or provide cross-ties at the top of your posts. (I like the gussets near the top so it has a “vaulted” appearance.)
Then you could install 2x6 decking on top of the beams. Install each section so the decking is parallel to the bottom edge.