Why were fortresses often built pentagonal?
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If we’re talking about bastion forts (which I guess, because pretty most pentagonal fortresses are bastion forts), aka star fortresses then pretty much everyone in this thread is wrong.
a) There were quite a few 4 sided bastion fortifications (like for example [Svartholm](https://upload.wikimedia.org/wikipedia/commons/9/94/Svartholma_from_air.jpg) in Finland). So you could totally build a 4 point bastion fort.
b) It’s not about supporting fire, because bastion forts with 4 points are just as capable of supporting each bastion as a 5 sided bastion fort (as long as all of the angles were correct). It’s about interior volume. Each bastion is a fairly exact distance from each other bastion where musket fire from one bastion is able to lay down effective enfilading fire to protect its neighbours (enfilading fire is guns firing along the side of the wall so that each bullet has the maximum hit chance if a whole bunch of soldiers are trying to scale the bastion). Then each bastion is joined by a curtain wall. So with the size of each bastion and curtain wall limited by how far supporting fire from neighbouring bastions could reach you had to create more bastions to get more interior space.
A very small bastion fort had 4 bastions joined by 4 curtain walls (although it probably had an outer fortification protecting the gate that formed a 5th point, like [Castillo de San marcos](https://cdn.britannica.com/68/95668-050-81424504/Castillo-de-San-Marcos-fortification-Spanish-St.jpg)).
As bastions increased in size they got more and more points, like [this fortified city at Naarden](https://home.kpn.nl/pagklein/images/20100521a.jpg) with 6 bastions. Italian cities with a bastion-fort-like outer wall could have dozens of bastions. Note the gun battery emplacements behind the “arrowhead” of each wall. Placing a gun battery in this position gave them a very useful field of fire (firing down the side of the wall) but putting them low and behind the high walls of the bastion arrowhead also meant that they could only receive return fire from a very narrow angle. So it was hard for the attacker to direct firepower at these gun batteries, while anyone trying to storm the fort would have to expose themselves to the defenders gunfire at close range (where they could fire grapeshot/canister shot. Which is basically turning a cannon into a big shotgun).
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