Sleep runs in cycles of roughly 90 minutes through light, deep and REM stages. Waking at the end of a cycle feels markedly better than being pulled out of the middle of one, which is why five full cycles can leave you fresher than six interrupted ones. Give a wake time and this works backwards to the bedtimes that land on a cycle boundary.
About 90 minutes on average, though it varies between people and lengthens through the night as the REM portion grows. The 90-minute figure is a useful approximation, not a fixed number, so treat the times here as a guide rather than a schedule to the minute.
Being woken from deep slow-wave sleep produces sleep inertia — the grogginess that can persist for half an hour or more. Waking at the end of a cycle, in light sleep, avoids most of it. This is the whole reason six and a half hours can beat seven.
Because falling asleep is not instant. Fifteen minutes is the usual estimate for a healthy adult. If you routinely take much longer, adjust it — and if it is consistently over half an hour, that is worth raising with a doctor rather than working around.
Seven to nine hours for most adults, which is five to six full cycles. Teenagers need eight to ten. The widely repeated idea that some people thrive on four hours applies to a very small fraction of the population, and most who believe it of themselves are simply used to being tired.