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Chutar vs. dar um chute vs. dar um pontapé

While chutar, dar um chute, and dar um pontapé all translate to to kick, the choice differs based on geography (Brazil vs. Portugal) and intention (sport, aggression, or guessing).

Chutar

A2
This is the most general verb for kicking. It is heavily used in sports like soccer and for moving objects with the foot. Uniquely, chutar is also used figuratively to mean to guess when you do not know an answer.
O jogador chutou a bola com força.
(The player kicked the ball with force.)
Eu não sabia a resposta, então chutei qualquer coisa.
(I did not know the answer, so I guessed anything.)
Pare de chutar a perna da cadeira.
(Stop kicking the leg of the chair.)
Ela chutou o balde e saiu do emprego.
(She kicked the bucket (gave up) and quit her job.)

Dar um chute

B1
This phrase focuses on the single act of producing a kick. It is often interchangeable with the physical meaning of chutar, but rarely used for the guessing meaning. It is very common in spoken Brazilian Portuguese.
O bebê começou a dar um chute na barriga da mãe.
(The baby started to kick inside the mother's belly.)
Ele tentou dar um chute na porta para abrir.
(He tried to give the door a kick to open it.)
O lutador de artes marciais foi dar um chute alto.
(The martial arts fighter went to deliver a high kick.)

Dar um pontapé

B2
Literally meaning to hit with the tip of the foot. In Brazilian Portuguese, dar um pontapé almost always implies aggression, violence, or a fight. However, in European Portuguese, it is the standard neutral way to say kick in non-sporting contexts.
Durante a briga, ele tentou dar um pontapé no outro homem.
(During the fight, he tried to kick the other man.)
O ladrão deu um pontapé na vitrine da loja.
(The thief kicked the shop window.)
Ele saiu da sala e deu um pontapé na parede de raiva.
(He left the room and kicked the wall in anger.)

Summary

Use chutar for sports, general object manipulation, or when guessing an answer. Use dar um chute to emphasize the specific action of kicking. Use dar um pontapé to describe violent or aggressive kicks in Brazil, or as a standard term for kicking outside of soccer context in Portugal.