The shooter, a 17-year-old senior, used a shotgun and intended to harm more than the student who was hit, Champaign County Sheriff Matthew Melvin said.
The shooter was in custody facing a preliminary charge of felonious assault, with an initial hearing scheduled Monday, authorities said.
"As tragic as this situation is, we are very, very fortunate," prosecutor Kevin Talebi said. "It could have been much worse. I'm very, very thankful that, as bad as it is, it didn't escalate to a much more tragic incident."
The West Liberty High School shooting, in a common hallway, was reported Friday morning just as classes were getting underway at the school complex, which is surrounded by farm fields 45 miles northwest of Columbus.
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Superintendent Kraig Hissong called Cole "a good kid." "He's a great student and a real positive person to have in our school system," Hissong said. "Our hearts are out with his family right now and with him as he's trying to recover from this."
Cole's family asked for prayers for him and for the shooter and his family.
"We are certain they have been deeply hurt as well," Cole's family said in a statement. "We are confident that God has a purpose and plan through this tragedy."
Senior Ashley Rabenstein said she was in class down the hall at the time. When students first heard an odd noise, they weren't sure what it might be because construction has been occurring at the property, she said.
She said her teacher checked the hallway, then ran back and said there was a shooter and threw desks against the door to block it. Students fled through the windows and ran through a cornfield to regroup at nearby houses, Rabenstein said.