Karageorge, 22, was found dead in a dumpster of an apparent self-inflicted gunshot wound. He was identified by tattoos at the scene, and a police spokesman said the investigation would continue with both the dumpster and the body of Karageorge being transported to the county morgue.
"At this time, there's a lot of questions that we're still trying to work out," Sgt. Rich Weiner told reporters at the scene. "But he was found inside of the dumpster, the handgun was found inside of the dumpster with him.
"At this time we are able to confirm through tattoos here at the scene that it is the body of Kosta Karageorge."
A police report says Karageorge's mother told authorities he has had concussions and spells of confusion. She said he texted a message Wednesday citing the concussions and saying he was sorry if he was "an embarrassment."
Karageorge was scheduled to be recognized on senior day during Saturday's home game against rival Michigan, but he had gone missing on Wednesday leading up to the 42-28 win by the No. 6 Buckeyes.
"The Ohio State University Department of Athletics was shocked and saddened to learn today of the death of student-athlete Kosta Karageorge, a senior from Columbus," the school said in a statement. "Our thoughts and prayers are with the Karageorge family, and those who knew him, during this most difficult time."
Karageorge had previously spent three seasons on the Ohio State wrestling team before walking on to help on the defensive line in August.
"I mean, I was friends with Kosta before he even joined the team," Ohio State left tackle Taylor Decker said Saturday after the game. "He's a great guy, and just for something like that to happen, it's obviously a shame.
"Coming from the wrestling team and coming to help us, you just really hope a guy like that is safe. Stuff like that is a lot bigger than football."
Information from The Associated Press was used in this report.