I would say no as it doesn't have a sustained roof primarily. The reason I say roof would be because some buildings only have posts and no walls.
The other reason is because a cirlce doesn't constitute a 'game room' as that's the new standard for measurement for area effects. Even though it's smaller, it's not, in itself, a 'game room'.
My two cents
If the circle was indoors, then you are trapping the game room. Where the trap body was determines where it will effect. If it's inside the circle, then the walls of the circle contain it. If outside of the circle, only effects outside of the circle.