Teagan comes from the Irish surname Tadhgán — meaning "little poet" or "little philosopher" (from tadhg, poet). The Welsh form means "beautiful."
Teagan entered the U.S. top 1000 as a girls' name in 2005 and has been climbing rapidly. The musician Tegan Quin (of Tegan and Sara) is a famous bearer of the alternate spelling. Today Teagan sits in the U.S. top 400.
Teagan does not currently appear in the US Social Security Administration's top 1,000 girls' names, so we don't publish a US rank or birth count for it. That says nothing about the name's standing elsewhere in the world — only that it sits outside the ranked US data we rely on.
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In Pythagorean numerology the letters of Teagan reduce to 3, The Communicator. This is a traditional interpretive system, not a factual claim about the name.