Shakespeare's King Lear is a perfect example of Aristotle's definition of a tragic hero. According to Aristotle's theory of tragedy, the protagonist should be of high birth—in this way, his or her downfall will be more effective, evoking pity and fear on the part of the audience. The audience sympathizes with the main character because the protagonist has committed a fatal mistake, which occurs as a result of their own character flaw. Aristotle calls the tragic flaw hamartia, and this is what leads to the hero's downfall.

King Lear decides to abdicate. But before leaving his kingdom to his three daughters, he wants to obtain evidence of how much they love him in order to decide which of his three children deserves the biggest portion of his legacy. King Lear expects a public declaration of his daughters' love, obedience, and gratitude. The two oldest daughters, Goneril and Regan, do everything they can in order to prove to their father how much they love him, whereas Cordelia, the youngest, does not try to please her father. She does not believe in hypocrisy and flattery, but rather in sincerity:

Unhappy that I am, I cannot heave
My heart into my mouth. I love your majesty
According to my bond; no more nor less. (act 1, scene 1)

Feeling humiliated by Cordelia's response, Lear renounces her and divides his kingdom between the other two daughters and their husbands. King Lear does not realize the difference between the sincere love of his daughter Cordelia and the untrue, exaggerated love shown by Goneril and Regan. This is his biggest mistake, and it's what causes his demise. Once the two daughters get what they want, they show their true colors by disrespecting and mistreating their father.

In the end, King Lear suffers for his mistake. While holding Cordelia's corpse in his arms, he realizes his tragic flaw: had he not been so easily fooled by Goneril and Regan on the account of his pride, he might have still had his kingdom, and his daughter might have still been alive. The play ends with Lear's death, making it the perfect tragic ending.

Additionally, the play itself is a great example of what Aristotle describes as a tragedy. There are clear causes and effects that drive the action. The play is mimetic, meaning that it imitates life; because of this, the audience is able to relate to the story more easily and is able to accomplish the catharsis of emotions (yet another important part of Aristotle's views on tragedy). The chorus, according to Aristotle, should be integrated to create atmosphere, echo an idea, comment on the action, and contribute to the plot. In King Lear, the role of the chorus is partially taken on by the Fool. This character observes the action and comments on it, often giving ironic and thought-provoking remarks.

King Lear does indeed display all the characteristics of a tragic hero as defined by Aristotle. He's a fundamentally decent character of noble birth brought low by a tragic flaw, and he experiences a reversal of fortune. Lear is of course a king, so that's a good start. His tragic flaw is his pride. The king asks each of his daughters to make public declarations of their love for him. When Cordelia refuses to play along with this tawdry charade, Lear angrily banishes her in a fit of pique. Lear's pride has been sorely injured by Cordelia and so he lashes out.

Lear's pride is much in evidence elsewhere. Despite having divided up his kingdom among his daughters, he still insists upon being treated like a king. He may have relinquished his kingdom but his pride demands that he receive the same level of respect as before. Yet Regan and Goneril are anything but respectful towards their father, treating him like a silly old fool. The end result of all this is a spectacular reversal of fortune. A great and noble king is reduced to a truly pathetic figure, a crazed outlaw wandering around on a stormy heath, railing angrily against the universe and its numerous iniquities. How the mighty have fallen.

Yes, King Lear does fit Aristotle's definition of a tragic hero. Aristotle stated a tragedy must be a drama about persons and things of some importance, where the highly placed hero is brought low through the combination of his or her own faults (the "tragic flaw") and external forces. The situation must be capable of being generalized, and it should induce pity and fear in the viewers. Finally, the drama must end with the attainment of understanding, bringing about a katharsis or "purging" that resolves the pity and fear the audience feels.

King Lear is a highly placed individual, and his proposal to retire from the cares of state is an affair of some importance. His "tragic flaw," the inability to distinguish between sincere and false devotion, is a common fault among people in general, and so audiences easily feel pity and fear when he falls victim to flattery and is ruthlessly betrayed by those he should have been able to trust the most. Finally, Lear approaches a state of understanding at the end of the play, when he realizes (though too late) that Cordelia was in fact the only one of his daughters to be true to him, and for the whole of his life he has been ignorant of the true nature of his role as a king and his own devotion to earthly vanity:

Come, let's away to prison:
We two alone will sing like birds i' the cage:
When thou dost ask me blessing, I'll kneel down,
And ask of thee forgiveness: so we'll live,
And pray, and sing, ... and we'll wear out,
In a walled prison, packs and sects of great ones,
That ebb and flow by the moon. (Act V, Scene 3)

Even though the play ends with the deaths of both Lear and Cordelia, these lines show that before his death, Lear had reached the complete comprehension (anagnorisis), the understanding and acceptance of how he truly fits into the scheme of things, which tragedy demands.

In so far as Lear is a king, he belongs to high rank/station, and therefore basically qualified in the role of a tragic hero. Again, he is generally inclined to goodness, and his passage from renunciation of kingship to his disaster & death shows a downward curve from prosperity to adsversity. And in this downfall, he arouses the emotions of pity and fear.

But the error which lies at the root of Lear's sufferings is his division of kingdom--Goneril & Regan being rewarded, and Cordelia banished--an error resulting from his 'anger'. It was a flaw more in the person/character of the old king than in the circumstances. This is not strictly the Aristotelian 'hamartia'. However, the tragedy of Lear assumes cosmic dimensions to become truly universal.

One needs to consider the idea of the "Great Chain of Being."  Lear is a King because the gods have deemed him worthy to be a king.  For the Elizabethan audience, the idea that a king would give his throne up without question is a crucial error.  The coronation of a monarch was as much a religious activity as it was a political one.  Shakespeare's  audience knew nothing good would come from Lear's decision.

To compound matters, Lear loses his temper and banishes his loving daughter and his most loyal Kent.  From then on, "that way madness lies," as Lear himself admits.

But the error, the cause, the decline begins before the play opens; Lear has already decided to divide his kingdom; that is the beginning of his doom.