To act is to do something, and to be active is to be busy doing things: to be moving around, playing or working. If you're an active person, you're full of life and energy, and you're always on the go.
Active animals aren't sleepy or lazy: they're busy moving around. And active kids do not like to sit still.
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If you have an active mind, it's always busy thinking. If you have an active schedule, it's very busy, packed with activities. And if a thing is active, it's doing something. Here's an active volcano.