The calculation, step by step
Starting from a bedtime, first add the estimated time it takes to actually fall asleep (commonly 15 minutes) — cycles only begin once you're asleep, not the moment you lie down. From that point, each candidate wake-up time is found by adding a whole number of 90-minute cycles: 3 cycles (4.5 hours of sleep), 4 cycles (6 hours), 5 cycles (7.5 hours), or 6 cycles (9 hours).
A worked example
Going to bed at 23:00, with a 15-minute fall-asleep buffer: the four suggested wake-up times are 03:45 (3 cycles, 4.5 hours of sleep), 05:15 (4 cycles, 6 hours), 06:45 (5 cycles, 7.5 hours), and 08:15 (6 cycles, 9 hours). Each option is exactly 90 minutes after the previous one, since each represents one additional complete sleep cycle.
A second worked example — an earlier bedtime
Going to bed 30 minutes earlier, at 22:30, with the same 15-minute buffer: the four options shift to 03:15, 04:45, 06:15, and 07:45 — each exactly 30 minutes earlier than the corresponding option from the 23:00 bedtime, since every option in the calculation shifts by the same amount as the bedtime itself.
Choosing among the four options
For most adults, the 5-cycle (7.5-hour) or 6-cycle (9-hour) options land closest to commonly recommended sleep-duration ranges — the shorter 3- and 4-cycle options are useful for unavoidable early mornings or genuine time constraints, not as a regular target. Picking whichever of the four options is closest to (but not later than) your actual required wake-up time, rather than always defaulting to the longest option regardless of schedule, is the practical way to use this calculation day-to-day.
Why this direction is useful specifically
This bedtime-to-wake-up direction answers "if I go to bed now, when should I plan to wake up" — useful when bedtime is already fixed (you're heading to bed right now) and you want to set an alarm that lands on a natural cycle break rather than an arbitrary round number of hours later. The reverse question — working backward from a required wake-up time to find a bedtime — is covered in a companion article on this site.