We would say enough space, but then, you would hold us guilty of generalization. So, follow these pointers:
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For gazebos meant for socializing (parties and gatherings of twelve to 25 guests; more would be a crowd): about fifteen feet on all sides, since your gazebo would be at the center.
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Gazebo for family and friends (twelve members): Corner or center units with an in-built space of about 16' x 16', and foreground of about 150 square feet.
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Intimate escapade: A very subjective consideration. Still, let it nestle amidst dense foliage, leaving about 10' space on all sides. And let your gazebo space be of the size of your bedroom.
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For meditation, reflection, yoga, workouts, and all other quiet moments of "your time": Follow the principle of minimum clutter, maximum space. Leave at least 10' open space on all sides.