The revelation of his own former humanity in Hellraiser II has resulted in the Cenobite, Pinhead (Doug Bradley), being split into two different entities: his former self, World War I British Army Captain Elliot Spencer, and a manifestation of Spencer's id, which takes on the form of Pinhead. While Spencer ends up in limbo, Pinhead is trapped, along with the puzzle box, amongst the writhing figures and distorted faces etched into the surface of an intricately carved pillar — the Pillar of Souls.
The pillar is bought by J.P. Monroe (Kevin Bernhardt), the rich owner of a popular nightclub called The Boiler Room. During her investigation, an ambitious young television reporter, Joey Summerskill (Terry Farrell), slowly begins to learn about Pinhead and the mysterious puzzle box. Joey is introduced to the pain the box can bring when she witnesses a teenage clubgoer being ripped apart by the box's chains in a hospital emergency room. Joey tracks the box and a young woman named Terri (Paula Marshall) to The Boiler Room nightclub. Terri had previously stolen the box from the nightclub.
Video tape interviews are recovered from the Channard Institute of one of Pinhead's former victims Kirsty Cotton (Ashley Laurence). Joey and Terri learn through the videos about the demonic Cenobites and the power of the Lament Configuration puzzle box and that it is the only means of sending Pinhead back to Hell. Pinhead remains dormant until one night several hooked chains shoot out of the pillar and rip into one of the club goers, Sandy (Aimée Leigh), whom Monroe had just recently slept with. After killing Sandy, Pinhead consumes her flesh and her distorted face appears on the pillar. Pinhead convinces Monroe to bring him more club members so he can feed on them and be freed from the pillar.
Meanwhile, Joey is contacted by the spirit of Elliot Spencer, who tells her that this "Pinhead" is a separate entity than the one encountered by Kirsty previously. Without Spencer's humanity to act as a balancing influence, this Pinhead is completely evil and has no sense of order. Rather than abide by the laws of the Cenobite realm, he will indiscriminately wreak havoc on Earth for his own pleasure unless he is stopped. In order to defeat him, Joey must reunite Spencer's spirit with Pinhead, fusing them back into a single entity.
Pinhead and Joey confront each other in The Boiler Room after Pinhead massacres the club's patrons in various grotesque ways. Pinhead says that he exists to force humanity to recognize the darkness in their heart and his freedom is the only way to enable it. He orders Joey to give him the box but she breaks free and flees the club. Pinhead resurrects the corpses of his victims as Cenobites; Terri, transformed into a cigarette-smoking Cenobite that can dream, which she was unable to do as a human; Monroe, who has pistons jabbed through his skull; the Barman, who has barbed wire wrapped around his head and spits fire from his mouth; The DJ, who has CDs impaled into his skull, and throws CDs full speed at his victims; and 'Doc', who has a TV camera forcibly embedded in his eye socket. Joey flees through the quiet streets, pursued by the new Cenobites. Local police are killed by the Cenobites as Joey enters a church and begs the priest to help her. Lacking in faith that demons could exist, the priest is appalled by the appearance of Pinhead. The Cenobites trap Joey on a construction site and prepare to torture her - but the Lament Configuration activates and they are quickly sent to hell. Joey finds herself in a heaven like realm and comes face to face with an apparition who appears to be her dead father. The apparition tells Joey to give him the Lament Configuration, aka the puzzle box, and is revealed to be Pinhead in disguise. Pinhead ensnares her in machinery and prepares to transform her into a Cenobite, but is confronted by Spencer's spirit, who forcibly fuses himself into Pinhead. Joey breaks free and uses the altered Lament Configuration to stab Pinhead through the heart, finally sending him back to Hell. With Pinhead's humanity restored, Joey buries the Puzzle Box in a pool of concrete at the construction site.
The final scene shows the finished product of the same site - a building built where Joey buried the box, with the interior design identical to the Lament Configuration.

Question: Where is Pinhead trapped?
Answer: The Pillar of Souls.
Alternative answer: in the surface of the Pillar of Souls