Coriolanus is a tragic hero because he meets all the criteria.
First, he is a man of elevated birth. He comes from one of the finest families in Rome. He proves himself as a hero and savior of Rome in his battles against the Volscian.
Like many tragic heroes before him, Coriolanus is guilty of hubris. His pride will not let him "lower" himself. He considers himself better than the plebeians who question him.
His excessive pride leads him to abandon Rome and his family and join the Volsci.
Once he turns his back on Rome, he is doomed. That he must die in the end is a foregone conclusion.
In the end, it is his pride that causes his downfall and that makes him a tragic hero.
Coriolanus is an excellent example of a tragic hero because his flaws bring about his destruction. He is unquestionably a heroic individual in the sense that he is self-sacrificing, brave, and an incredible fighter. He gives his fellow soldiers a rousing speech about battle:
If any think brave death outweighs bad life
And that his country's dearer than himself;
Let him alone, or so many so minded,
Wave thus, to express his disposition,
And follow Marcius [Coriolanus].
In a remarkable feat, Coriolanus almost single-handedly storms Corioli, the town from which he gets his name. Other positive qualities include a sense of honesty. He despises lies. When his mother Volumnia asks him to temper his rhetoric, he says, “would you have me / False to my nature? Rather say I play / The man I am.”
However, the titular Coriolanus also has numerous faults. As mentioned, he has a severe case of pride, or what is perceived to be pride. He abhors the common people and believes that the patricians have a natural right to rule the lower classes. Coriolanus can also be extremely unbending, a quality that serves him well in battle where he must be fearless and resolute. In a political and social context, however, this inflexibility does not allow him to work with others.
After the plebeians banish Coriolanus from Rome because of his aggression and unwillingness to play the political game, he turns against the city, joins his enemies, and threatens Rome’s very existence. It is no surprise that he ends up dying at the hands of his former enemies Aufidius and the Volsces, the very people from whom he took Corioli. Ultimately, his flaws make him unable to fit in both with the Romans and the Volsces, leading to his death. In spite of Coriolanus’s destructiveness, Aufidius asserts “Yet he [Coriolanus] shall have a noble memory.” He is indeed a tragic hero.