I'm not sure what you mean by "standing right behind the circle", but the answer is probably no.
A full 180 degree view is called "fish eye" and, as its name suggests there is some distortion.
Consumer priced camcorders have zoom lenses. Some have optional adapters to make the wide setting wider, but not enough if you are near the sideline.
The widest view in a common camcorder is probably the GoPro, so you might look at that. It is wide because it is made to be worn or mounted and not precisely aimed.
The only way to get full 180 degree viewing fields on video that I can think of is with a camera where you can change lenses, like a DSLR and mount a "fish eye" lens on it.
If you stand back far enough to get a common consumer camera or camcorder to work, the people on the sidelines will be in the way. I've seen people use long poles with their camera mounted on top just to get a unique perspective. If I really wanted the shot your after, I would use my camera to find out how far back I would need to stand with it set at its widest view. Then I would attach it to a ten foot piece of plastic pipe, start the recording and hold it up so it shoots over the crowd.