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Why do Jewish months have 29 or 30 days?

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Circular diagram of the Jewish religious year showing Hebrew months around the outer ring and Gregorian months inside, with major holidays and fasts such as Rosh Hashanah, Yom Kippur, Sukkot, Hanukkah, Passover, and Purim labeled with their dates. — Jewish religious year The months and holidays of the Jewish religious year.

The Jewish calendar is lunisolar, meaning it is regulated by the cycles of both the moon and the sun. Because the synodic month—the average interval between two new moons—is approximately 29 days, 12 hours, and 44 minutes, the calendar must account for these fractional parts using only complete days.12

To reconcile this, the calendar uses a schematic system of "full" and "defective" months:2

Full months contain 30 days.2
Defective months contain 29 days.2

While most months have a fixed length, the months of Cheshvan and Kislev vary between 29 and 30 days depending on the specific requirements of the year to ensure that religious festivals fall on the appropriate days of the week. This system allows the calendar to remain independent of the true new moon while maintaining the necessary structure for the Jewish religious year.12