How Deep Does a Cut Have to Be Before Going to the Hospital?

Over the weekend I sliced my thumb. It wasn't just a paper cut, but a deep slice causing a massive downpour of blood. However, I didn't think it was enough to actually see a doctor so I treated it as I treat any wound. I just ran my thumb under cold water, slapped on a band-aid and put an ice pack on it for a while.

My wife asked if I should go to the hospital, but I assured her it was fine. That got me thinking - at what point should you go to a hospital?

Now, it was deep, but not to the bone. Also, if I would've cut much further, it would've sliced the tip of my thumb clean off. In that case, yes, I would've seen a doctor. I thought about super gluing it together but I figured if I just keep it clean and pressed it would heal itself, which it has.

What about you? Have you gone to the doctor for less? Let me know in the comments.