I was curious about what was used in zoos etc today, apparently the San Diego Zoo has some of the best aviaries in the world, they were able to enclose natural canyons to create really large enclosed areas. One of the biggest being about 25m high, 21m wide and 46m long.
Give your park the right natural geography and I imagine you could quite believably build an enclosed aviary that dwarfs that one
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Then as overread said it comes down to the temperment and natural behaviour of the animals rather then just physical excercise.