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Copo vs. xícara vs. chávena

In Portuguese, the words copo, xícara, and chávena all refer to vessels for drinking, but they are not interchangeable. The choice of word depends on the type of beverage (hot or cold), the material of the vessel, and even the region where the language is spoken.

Copo

A1
A copo is the general term for a drinking glass or a simple cup, typically used for cold beverages. It is usually made of glass or plastic and does not have a handle.
Pode me trazer um copo de água, por favor?
(Can you bring me a glass of water, please?)
Ele encheu o copo com sumo de laranja.
(He filled the glass with orange juice.)
Na festa, havia copos de plástico para a cerveja.
(At the party, there were only plastic cups for the beer.)
Cuidado para não quebrar o copo de vidro.
(Be careful not to break the glass cup.)
A receita pede meio copo de óleo.
(The recipe asks for half a cup of oil.)

Xícara

A1
A xícara is a cup, almost always with a handle and often accompanied by a saucer (pires). It is typically made of ceramic or porcelain and is used for hot drinks like coffee or tea. This is the standard term used in Brazilian Portuguese.
Gosto de tomar uma xícara de café depois do almoço.
(I like to have a cup of coffee after lunch.)
Ela me serviu uma xícara de chá de camomila.
(She served me a cup of chamomile tea.)
Para o bolo, use duas xícaras de farinha.
(For the cake, use two cups of flour.)
Quebrei a asa da minha xícara favorita.
(I broke the handle of my favorite cup.)
Ele coleciona xícaras de diferentes países.
(He collects cups from different countries.)

Chávena

B1
A chávena is functionally identical to a xícara: a small cup with a handle and saucer, used for hot beverages and made of porcelain or ceramic. The main difference is regional, as chávena is the preferred term in European Portuguese, while xícara is almost exclusively used in Brazil.
Em Portugal, pede-se uma chávena de café.
(In Portugal, you ask for a cup of coffee.)
A minha avó ofereceu-me um conjunto de chávenas de chá.
(My grandmother gave me a set of teacups.)
Podes aquecer o leite numa chávena no micro-ondas?
(Can you heat the milk in a cup in the microwave?)
A chávena escorregou e partiu-se no chão.
(The cup slipped and broke on the floor.)
Esta chávena de porcelana é muito delicada.
(This porcelain cup is very delicate.)

Summary

In summary, use copo for a glass or simple cup for cold drinks like water, juice, or soda. Use xícara for a cup with a handle for hot drinks like coffee and tea if you are speaking Brazilian Portuguese. Use chávena for the exact same type of cup (for hot drinks, with a handle) if you are speaking European Portuguese. The key distinctions are the beverage's temperature and regional dialect.