We don't have a specific measurement. The description just says 'a tall hat.' It would appear to be about 12 to 15 inches.
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Mostly long grey robes with a silver belt. A tall pointed blue hat, a long grey cloak, a silver scarf over which a white beard hung down below his waist, and immense black boots. A staff that looked like twisted vines.
Frodo knows Gandalf because Gandalf was like Bilbo (Frodo's grand-uncle) guardian on Bilbo's adventures. Now, Frodo knows Gandalf through Bilbo.
He came without his coat and hat, so a cloak and hood were loaned to him. He left without his handkerchief and borrowed some from Elrond. And Gandalf arrived with his keys.
Gandalf was known by different names to different peoples: Gandalf to Northern men, Mithrandir to the elves and Dunedain (as in Gondor), and Tharkun to the Dwarves. Olorin was his true name as a Maia.
The hobbits of the Shire might describe describe Gandalf as wise, powerful, and fun-filled.(Gandalf is over 3000 years old!)