tuk,
If the structure is to support thin-film material, you can build with fairly-delicate material, so long as it doesn't bend. Note that tent frames use fiberglass rod frames, which are thin, relative to length. The weight of the struts, themselves, will likely outweigh any film/fabric covering
The bottom-line is how much weight the geodesic frame has to support, and how much wind-load it must withstand. Perhaps, snow-load is not a consideration, but rain-wetted fabric can get heavy, at the size you propose.
I've seen dome frames of this size, made of bent bamboo, so, pretty light struts.
I've encountered poor & good quality lumber, from the building supply store, so, I know that there is variability in the density & strength of lumber of the same dimensions. Since I don't know how hard/soft is the wood you have available, I shouldn't comment on the dimensions you have described. (I also don't know how dry/wet is the wood.)
The lengths of your proposed struts are fairly short (relative to cross-section,) and that's in your favor.
You could build a test section, and test the strength, yourself. See how much it will support, without deformation. Continue adding weight, until it does fail. (This is a common approach to assessing construction strength.)
You should note that joint-connections will likely fail before there is any problem with the length of a strut. Joints/hubs must withstand compression (which is generally not a problem,) as well as tension. As weight is added to a dome structure, the joints around the "equator" will be under the increasing tension. This is why you will often see metal straps added to horizontal, and near-horizontal, strut joints. Personally, I would not trust nails/screws only. You should, at least, include a disk or ring to each joint, on the outside, or inside, or both. (See attached image.) You can see simple joint reinforcements (and all sorts of hubs) at:
http://www.domerama.com/dome-basics/geodesic-dome-hub-connectors/
-Taff