Cerco vs. sítio
While both cerco and sítio can be translated as siege in Portuguese, they have different nuances. Cerco is the most common and direct term, emphasizing the action of surrounding, while sítio is a more historical or formal term and has other very common meanings.
Cerco
B2Cerco is the most frequent and direct word for siege. It comes from the verb cercar (to surround) and focuses on the military action of surrounding and blockading a location. It is the standard term for modern military or police operations and is also used figuratively.
O cerco de Constantinopla foi um ponto de viragem na história.
(The siege of Constantinople was a turning point in history.)
A polícia montou um cerco ao prédio onde o suspeito se escondeu.
(The police set up a siege around the building where the suspect was hiding.)
A cidade preparou-se para um longo cerco, armazenando comida e água.
(The city prepared for a long siege, storing food and water.)
Com as novas evidências, o cerco ao político corrupto apertou-se.
(With the new evidence, the net closed in on the corrupt politician.)
Sítio
A2Sítio can also mean siege, but this usage is more formal, literary, or historical. Its primary modern meanings are place, site, or small farm (especially in Brazil), so context is crucial. As a siege, it often refers to the state of being besieged. The related verb sitiar (to besiege) is quite common.
O famoso Sítio de Troia é contado na Ilíada de Homero.
(The famous Siege of Troy is told in Homer's Iliad.)
As crónicas medievais descrevem o longo sítio ao castelo.
(The medieval chronicles describe the long siege of the castle.)
A população sofreu muito durante os três anos de sítio.
(The population suffered greatly during the three years of siege.)
Eles passaram o fim de semana no seu sítio no campo. (Shows the more common meaning)
(They spent the weekend at their small farm in the countryside.)
Summary
In short, use cerco for the common, modern-day meaning of a military or police siege, as it clearly refers to the action of surrounding. Use sítio for a siege in more historical, formal, or literary contexts. Remember that sítio most commonly means place or small farm, making cerco the less ambiguous choice for siege in general conversation.