What time should I go to bed to wake up at 10:00 am?

Between 12:45 am and 2:45 am, which allows 15 minutes to fall asleep and gives you seven to nine hours.

How many sleep cycles is that?

Five ninety-minute cycles is 7 hours 30 minutes, putting sleep onset at 2:30 am. Six is 9 hours, at 1:00 am.

Does waking at the end of a cycle actually help?

It is a reasonable heuristic and not a schedule. Cycle length varies from about 70 to 120 minutes between people and across a single night, so a calculator cannot know where your boundaries fall. Total sleep matters more than the minute you wake on.

What time should I go to sleep to wake up at 10:00 am?

To wake at 10:00 am on seven to nine hours

Be asleep between 1:00 am and 3:00 am

In bed between 12:45 am and 2:45 am, allowing 15 minutes to fall asleep

Seven hours is the floor the American Academy of Sleep Medicine's consensus statement sets for adults, so to wake at 10:00 am you want to be asleep by 3:00 am at the latest. Nine hours puts the other end at 1:00 am. Neither is a cliff edge; they bracket the range where most adults are getting enough.

By the hour

All of these allow 15 minutes to fall asleep, which is the middle of the normal range. Consistently taking under five minutes usually means you are short on sleep; consistently taking more than half an hour is the definition used for sleep-onset insomnia.
Sleep	Asleep by	In bed by
9 hours	1:00 am	12:45 am
8.5 hours	1:30 am	1:15 am
8 hours	2:00 am	1:45 am
7.5 hours	2:30 am	2:15 am
7 hours	3:00 am	2:45 am
6.5 hours	3:30 am	3:15 am
6 hours	4:00 am	3:45 am

By sleep cycle

Cycles	Time asleep	When to be asleep
6 cycles	9 hours	Asleep at 1:00 am — in bed at 12:45 am
5 cycles	7 hours 30 minutes	Asleep at 2:30 am — in bed at 2:15 am
4 cycles	6 hours	Asleep at 4:00 am — in bed at 3:45 am
3 cycles	4 hours 30 minutes	Asleep at 5:30 am — in bed at 5:15 am

The cycle-aligned times below assume a flat 90-minute cycle, which is a population average rather than a fact about any one night. Real cycles run about 70 to 120 minutes, the first ones are shorter than the later ones, and their composition shifts from deep sleep towards REM as the night goes on. Treat the numbers as a target to aim at, not a schedule to hit. Total sleep matters considerably more than which minute you wake on.

What this actually assumes

That you fall asleep in about 15 minutes. If you routinely lie awake for 45, every time above moves earlier by half an hour.
That your cycles are 90 minutes. Across people they run 70 to 120, so by the fourth cycle the accumulated error can exceed an hour.
That you sleep through. Brief awakenings are normal and get more frequent with age; they do not reset the cycle count but they do add to the time in bed.
That the seven-hour floor applies to you. It is an adult figure. Teenagers need eight to ten and children more again.

How much sleep you need

Age	Per day	Notes
Babies, 4 to 12 months	12 to 16 hours, including naps	Counting naps. Sleep at this age is still spread across the day and the total is what matters, not the shape of it.
Toddlers, 1 to 2 years	11 to 14 hours, including naps	Counting naps. Most children drop to a single daytime nap somewhere in this band.
Preschoolers, 3 to 5 years	10 to 13 hours, including naps	Counting naps, which most children stop taking towards the end of this band.
Children, 6 to 12 years	9 to 12 hours	Naps have usually stopped by now, so this is night sleep.
Teenagers, 13 to 18 years	8 to 10 hours	Adolescent circadian timing shifts later by one to three hours at puberty, which is a biological change and not a behavioural one. A teenager who cannot fall asleep at 10 pm is usually not being difficult, and an early school start subtracts from the wrong end of the night.
Adults, 18 and over	7 or more hours	The consensus statement gives a floor, not a range: seven or more hours on a regular basis. It does not endorse eight as a target, and it declines to state an upper bound, because regularly sleeping more than nine hours is associated with poor health outcomes without the direction of causation being established.

Common questions

What time should I go to bed to wake up at 10:00 am?: Between 12:45 am and 2:45 am, which allows 15 minutes to fall asleep and gives you seven to nine hours.
How many sleep cycles is that?: Five ninety-minute cycles is 7 hours 30 minutes, putting sleep onset at 2:30 am. Six is 9 hours, at 1:00 am.
Does waking at the end of a cycle actually help?: It is a reasonable heuristic and not a schedule. Cycle length varies from about 70 to 120 minutes between people and across a single night, so a calculator cannot know where your boundaries fall. Total sleep matters more than the minute you wake on.

Waking at 9:00 am Asleep by 2:00 am for seven hours
Waking at 9:30 am Asleep by 2:30 am for seven hours
Waking at 10:30 am Asleep by 3:30 am for seven hours
Waking at 11:00 am Asleep by 4:00 am for seven hours
Going to bed at 2:45 am The same question from the other end
What a sleep cycle actually is And why the 90-minute figure is an average, not a clock
When to stop drinking coffee Counted back from a 10:00 am wake-up

Sources

Recommended Amount of Sleep for a Healthy Adult: A Joint Consensus Statement — Watson et al., Sleep 38(6) / Journal of Clinical Sleep Medicine, 2015
Normal Human Sleep: An Overview — Carskadon & Dement, Principles and Practice of Sleep Medicine, 6th ed., 2017

How these figures are compiled and checked