A stalactite is a type of formation that hangs from the ceiling of caves, hot springs, or even structures such as bridges and mines.  Whereas, a stalagmite is a formation that occurs when stalactite drip to the ground, creating columns, rising from the floor of a cave. They often unite with the stalactite.  Cave formations take millions of years to form.

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Easy to know the difference by remembering a stalactite holds 'tight' to the ceiling and a stalagmite grows 'mighty' from the ground.