花甲 refers to the 六十甲子 — the sixty-year cycle formed by pairing the ten Heavenly Stems (天干) with the twelve Earthly Branches (地支). One full cycle = sixty years, so 花甲之年 = age 60. Related milestones: 而立 30, 不惑 40, 知天命 50, 花甲 60, 古稀 70.
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Formal and respectful. Use for elders or in writing; colloquially people just say 六十岁 or 六十多岁.