Round pens, known as "breaking pens" or "bull pens", are often only 35' in diameter. These tiny pens can be useful for slow unmounted work - for instance, teaching a horse to tie or lead, sacking out a horse, introducing a horse to tack. They can also be useful for teaching horses and new owners the first basics of roundpenning at a walk and jog, and they can be useful places to turn out a stallion or a foal to get some air and more freedom and exercise than they could manage to get in a stall, but that's about the extent of their usefulness. Even if the footing is perfect and your horse is warmed up very well before you take him into the round pen, the tiny circles are just too stressful for leg bones and joints. This is not a pen to use for fast exercise or serious lunging - unless the horse is highly trained and superbly developed physically, it won't be able to move correctly or even carry itself correctly bent on such a small circle. And it's definitely not a suitable size for ridden work. If the horse is carrying a rider, there are even more demands placed on the horse's legs and balance, and the size of the pen becomes even more critical.

A "working pen" might be 50' or more in diameter. This would allow you to work a horse at a walk, jog, and trot - but probably not at a canter, unless the horse was compact, balanced, experienced, and well-trained... and it would definitely NOT be the place to introduce a horse to the idea of cantering under a rider.

A "training pen" - this is the point at which a round pen becomes REALLY useful - would be at least 66' in diameter.

A 20-meter circle - the basic working circle for a low-level dressage horse in the first years of its training, and a good MINIMUM size of circle for any horse at the start of its training - would be 66' across, and since horses don't usually work up against the side of the pen, but on a track to the inside of the pen wall, I would suggest that any round pen used for riding be at least 70' in diameter. If you're going to do a lot of cantering in the pen, it wouldn't hurt to make it even larger. The size of the pen needs to correspond to the sort of work you plan to do in it. A pen 75' in diameter would be ridiculously large for someone who planned to use the pen only for sacking out and tying lessons. On the other hand, pens used by trainers who do cow work are even larger - sometimes four or five times that size.