The simplest imaginable tent design, therefore, would be just a center pole and a bunch of fabric hanging from it, staked to the ground some distance away so there's space inside. Unfortunately, to get any significant room inside the tent you need either an enormously tall center pole (hard to transport and pitch, and vulnerable to wind) or stakes a mile away (wasting a lot of real estate on a portion of the tent that's too close to the ground to use).
One solution to this is to use two or more center poles, often with a ridge pole to hold the shape of the roof in between them. When I was growing up, this design was used for "pup tents" and low-end backpacking tents. The result has triangular ends and rectangular (or slightly trapezoidal) sides at a diagonal (so they serve as both wall and roof).
This shape appears in a number of sources, including King René and a 15th-century Italian treatise on military engineering, De machinis by Paolo Santini (Lat. ms. 7229, Bib. Nat. Paris). Pages 13v-14r of the latter source shows a military camp with twelve circular tents like our first King René tent, "pup tents", and one walled rectangle (looking like a pup tent on top of vertical walls). Bear in mind that this shape can also be attained by an A-frame construction (see below), and in many of the pictures it's hard to tell which design is used.
The Norse Film and Pageant Society has reconstructed such a thing,
calling it the Anglo-Saxon
Geteld; I don't know what sources they worked from.
Cariadoc and Elizabeth (David Friedman and Betty Cook)
have likewise reconstructed one
based on a 9th-century picture, and it seems to have worked well at
Pennsic. Here are two pictures.
(The larger tent housed two children, Rebecca and Bill, while the tiny tent, made from the larger one's scraps, housed their stuffed animals.)
Another interpretation of the geteld is due to Thora Sharptooth and Dofinn-Hallr Morrisson (Carolyn and Greg Priest-Dorman). The sources they examined uniformly showed the ridgepole extending beyond the end the end of the roof, and running not under the roof, but through a sleeve above it formed, apparently, by extending the roof fabric. They're working on a full-fledged Web page about the project, but this picture will have to do for now.
Meghan Roberts (aka Magnunnr Hringsdottir) has built a smaller geteld,
of wool fabric with hand-sewn seams. She decided to have the
"vertical" posts angled in a little, forming a trapezoid
rather than a rectangle, for stability.
Click on the picture to see her site with more details on the tent.
Note: the picture at left doesn't show sleeves extending beyond the ends of the roof, but she's adding that.
This approach has several disadvantages. One practical disadvantage is transporting all those side poles. Another is that unless the roof is steeply pitched, rainwater can collect in pools in the vicinity of the side poles; a roof that sheds raindrops nicely may leak like a sieve when rainwater pools on it for minutes or hours. These may explain the third disadvantage: as David Kuijt (Dafydd ap Gwystl) points out, is that there is essentially no evidence that side poles were ever used in medieval European tents.
This approach is taken in a surviving
16th-century (?) round tent. Note that in this picture, it's been
retrofitted with an internal structure to set it up inside a museum, but
originally it was just a single center-pole and guy-ropes. Note also
that the guy ropes attach to the shoulder, rather than going through
the shoulder and up to the peak (as in our King René tents).
My thanks to Terafan Greydragon for finding this surviving tent in a
museum in Basel, Switzerland, photographing it, and Webbing it; thanks
also to Tanya Guptill for pointing it out to me.
On the left side of this picture is another Viking A-frame; behind it is a variant, invented by Cariadoc as far as I know, in which one of the walls can be held out by another pair of beams to form an awning (which, incidentally, must be lowered when it rains, or it'll collect water and leak like a sieve). This awning-tent has stored kitchen supplies in Enchanted Ground for a number of years.
Bear in mind, when looking at surviving pictures of triangular tents,
that this effect can be achieved with either an A-frame or a "pup tent"
design (see above).