Why do 12 lunar months not equal one tropical year?

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Kalendarium (“Calendar”) by Regiomontanus — calendar First complete printed title page for the *Kalendarium* (“Calendar”) by Regiomontanus, 1476.

Twelve lunar months do not equal one tropical year because the two cycles are incommensurable, meaning they do not align perfectly in length.1

A synodic month—the time required for the Moon to complete a full cycle of phases—averages approximately 29.53 days. Consequently, 12 synodic months total about 354.37 days. In contrast, a tropical year, which tracks the Earth's orbit around the Sun relative to the equinoxes, is approximately 365.24 days long.213

This discrepancy leaves the lunar-based year roughly 11 days shorter than the solar-based year. Because of this difference, calendars that attempt to remain in step with both the Moon's phases and the seasons must use intercalation—the periodic addition of days or entire months—to reconcile the two cycles.14