Waking at 3:00 am is a the small hours alarm, so the bedtimes are unconventional: 5:46 pm for six cycles, 7:16 pm for five, 8:46 pm for four. These are shift patterns rather than ordinary evenings.
Why work backwards from the alarm
Bedtime is the variable you control. The alarm is set by work, school or a train and does not move. So the useful question is how late you can stay up and still get whole cycles — which is what this page answers.
How this is worked out
Sleep runs in cycles of roughly 90 minutes, passing from light sleep into deep sleep and REM and back out. Waking at the end of one means waking out of light sleep. We count whole cycles back from your alarm and then subtract 14 minutes for the time it takes to fall asleep, which is why these times are not round numbers.
Limitations
Ninety minutes is an average, not your number. Real cycles run about 70 to 120 minutes and lengthen through the night, so the six-cycle time is a much looser target than the four-cycle one. Treat any of these as a window of about three quarters of an hour rather than a time to the minute. The 14-minute allowance for falling asleep is the same kind of average; if you routinely take half an hour, every time on this page is sixteen minutes out until you change it on the sleep cycle calculator.
Total sleep matters more than boundary timing. Do not trade an hour of sleep for a tidier alarm, and do not stay up to reach a cycle boundary you have already missed.
This page does arithmetic on a wall clock. It does not measure your sleep, it cannot tell you how much you need, and it is not medical advice. Persistent tiredness that a better bedtime does not fix — insomnia, sleep apnoea, restless legs, shift-work disorder — is common, treatable, and belongs with a doctor rather than with a calculator.