One thing I dislike about teach geometry are the number of theorems and how old school geometry is in the way of teaching it. I know I hear it coming already about patty paper and all the things you can do with it, but it's not the same. There is no real life situation when you have to know the Central angle theorem.
The way the game is played is everyone is assigned a number and stands in a circle, one person throws the ball up into the air and calls out a number, when that person touches it they yell stop. All players stop and they get 2 free jumps to get closer. If they get hit, they are out, if it is caught or misses the thrower is out.
The way I mathematized it was at the beginning of the round they had to name the polygon they formed if someone was out and named it first they got to jump back in. I played it with two groups outside and worked well since there were shortened periods.