Fan-shaped weapons were used in Japan. There were 5 main types, one of which was the folding fan.
To begin with the folding fan, the weaponised versions were just folding fans with the outer ribs made of iron (or sometimes bronze/brass). The inner ribs were usually bamboo or wood, but sometimes iron. These fans were called tessen, "iron fan", or menbarigata tessen or sensugata tessen [1] to distinguish them from other types of tessen. These were not weapons of choice, but emergency self-defence weapons. These have two advantages over many other weapons:
The fan was a common dress accessory for samurai (and upper-class women), and could be worn where weapons were not allowed or appropriate. Thus, a reinforced fighting-ready fan could be at hand at all times.
It was common and socially acceptable to carry the fan in-hand, so it would be ready for parrying a surprise attack (after which, a sword could be drawn).
The fan would usually be used as an emergency parrying device, or as a club. It would be kept folded when used as a weapon, not opened out as seen in some movies.
These could, if not too heavy, still be used as fans.
Two examples:
Iron inner ribs: https://commons.wikimedia.org/wiki/File:Antique_Japanese_(samurai)_gunsen_fan_6.JPG
Bamboo inner ribs: https://commons.wikimedia.org/wiki/File:Japanese_gunsen_fan_6.JPG
There were also solid iron (or bronze) tessen. The most common were in the shape of a folded folding fan. These were much heavier than the folding type (and of course couldn't be used as fans). These varied from quite detailed, with simulated folds along the sides, to others that are fan-shaped, but clearly iron truncheons. These solid tessen are often called tenarashi tessen, "hand-training iron fan", which suggests that one use of these was as a strength-training tool rather than being intended as a disguised self-defence weapon. Some examples:
A third type of fan used in fighting was primarily a signalling device, the gunbai. This was a permanently-open fan, used on the battlefield to visually convey orders, and were carried by commanders. These were often quite large, for better visibility. An example:
These were sometimes strongly built, in which case they could be used to parry attacks. It is said that Takeda Shingen resorted to his gunbai to defend himself during a battle in 1561, when an enemy cavalryman reached his command position (Mol, 2003).
A fourth type is the combination of the previous two: a gunbai-shaped tenarashi tessen. These are often somewhat larger and heavier than the "folding" type of tenarashi tessen, but smaller than real gunbai (since otherwise they would be monstrously heavy).
Finally, there were tanto (knives/daggers) disguised as fans. These typically had wooden hilts and scabbards carved and painted to look like folded folding fans:
Most of the existing fan-tanto appear to be relatively recently made replicas or tourist souvenirs. These would be used simply as knives/daggers, rather than parrying devices or truncheons like the other types of fan weapons. Sometimes, tenarashi tessen were made with hidden blades (but these are rare); these offered the opportunity for use in all of these modes.
The martial art of fighting with the tessen, tessen-jutsu, was taught by some jujutsu and kenjutsu schools, but it was only regarded as a minor auxiliary weapon. There are a small number of styles that specialise in tessen-jutsu, but these appear to be recent (Mol, 2003).
Cunningham (2015) and Mol (2003) discuss the tessen and tessen-jutsu. These books are also very useful for providing context for the role of the tessen, since they discuss many other minor weapons, hidden weapons, disguised weapons, and "secret" weapons.
Endnotes and references:
[1] I can't at the moment easily look up what these mean, but my provisional translations are "outer rib shaped iron fan" and "men's fan shaped iron fan", respectively.
Don Cunningham, Secret Weapons of Jujutsu, Tuttle, 2015.
Serge Mol, Classical Weaponry of Japan: Special Weapons and Tactics of the Martial Arts, Kodansha, 2003.