What do the horse people call...

https://en.wikipedia.org/wiki/Round_pen

Perfect. Thanks!

I work in horsey country

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"bull ring"

Yes, Bull-ring is the term that I have heard.
Father of a horsey girl.

Arena

It's called a round pen here when used for horses. If it's for cattle it's called the bull pen or arena.

Thunder Dome?

James Fleming, post: 449721, member: 136 wrote: I work in horsey country

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Do you wear the hat and blow a bugle? When I was a lot younger I used to chase foxes too.

Hack, post: 449918, member: 708 wrote: Do you wear the hat and blow a bugle?

Nope...just survey their multi million dollar farms. Plus I'm a little above my former riding weight, I'd look out of place on a draft horse.