The play is set in Troezen, a coastal town in the northeastern Peloponnese. Theseus, the king of Athens, is serving a year's voluntary exile after having murdered a local king and his sons. His illegitimate son is Hippolytus, whose birth is the result of Theseus's rape of the Amazon Hippolyta. Hippolytus has been trained since childhood by the king of Troezen, Pittheus.
At the opening of the play Aphrodite, Goddess of love, explains that Hippolytus has sworn chastity and refuses to revere her. Instead, he honors the Goddess of the hunt, Artemis. This has led her to initiate a plan of vengeance on Hippolytus. When Hippolytus went to Athens two years previously Aphrodite inspired Phaedra, Hippolytus' stepmother, to fall in love with him.
Hippolytus appears with his followers and shows reverence to a statue of Artemis, a chaste goddess. A servant warns him about slighting Aphrodite, but Hippolytus refuses to listen.
The chorus, consisting of young married women of Troezen, enters and describes how Theseus's wife, Phaedra has not eaten or slept in three days. Phaedra, sickly, appears with her nurse. After an agonizing discussion, Phaedra finally confesses why she is ill: she loves Hippolytus. The nurse and the chorus are shocked. Phaedra explains that she must starve herself and die with her honor intact. However, the nurse quickly retracts her initial response and tells Phaedra that she has a magical charm to cure her. However, in an aside she reveals different plans.
The nurse, after making Hippolytus swear not to tell anyone, informs Hippolytus of Phaedra's desire and suggests that Hippolytus consider yielding to her. He reacts with a furious tirade and threatens to tell his father, Theseus, everything as soon as he arrives. Phaedra realizes disaster has fallen. After making the chorus swear secrecy, she goes inside and hangs herself.
Theseus returns and discovers his wife's dead body. Because the chorus is sworn to secrecy, they cannot tell Theseus why she killed herself. Theseus discovers a letter on Phaedra's body, which falsely asserts that she was raped by Hippolytus. Enraged, Theseus curses his son either to death or at least exile. To execute the curse, Theseus calls upon his father, the god Poseidon, who has promised to grant his son three wishes. Hippolytus enters and protests his innocence but cannot tell the truth because of the binding oath that he swore. Taking his wife's letter as proof, Theseus exiles his son.
The chorus sings a lament for Hippolytus.
A messenger enters and describes a gruesome scene to Theseus; as Hippolytus got in his chariot to leave the kingdom, a bull roared out of the sea, frightening his horses, which dashed his chariot among the rocks, dragging Hippolytus behind. Hippolytus seems to be dying. The messenger protests Hippolytus' innocence, but Theseus refuses to believe him.
Theseus is glad that Hippolytus is suffering and about to die. But then the goddess, Artemis, appears and rages at Theseus for killing his own son; she brutally tells him the truth: there was no rape, Phaedra had lied, his son was innocent. Theseus is painfully devastated by this revelation. Hippolytus is carried in physically battered and barely clinging to life. In the last moments of the play, Hippolytus forgives his father, kind words are exchanged between father and son, and then Hippolytus dies.

Question: What has Hippolytus sworn?
Answer: Chastity.
Alternative answer: not to tell Phaedra's secret love of him