Are sleep cycles really ninety minutes?

On average, roughly. In practice they run somewhere between 70 and 120 minutes and change through the night, getting longer towards morning. So these times are approximate by construction, and no calculator can tell you where your cycles actually fall on a given night.

Is it better to hit a boundary or get more sleep?

More sleep, nearly always. Six cycles is nine hours and four is six, and waking cleanly on too little is worse than waking a bit groggy on enough. The boundary idea is worth something when you are choosing between two reasonable bedtimes, and not worth cutting an hour for.

Why add time to fall asleep?

Because going to bed and being asleep are not the same moment, and the gap is typically ten to twenty minutes. Leaving it out shifts every answer by that much, which is enough to land you in the middle of a cycle rather than between two.

Sleep Calculator

When to go to bed.

Runs in your browserNothing you enter is uploaded, and it keeps working with the network off.

I know

I have to wake atI am going to bed now

Time

24 hour clock.

Minutes to fall asleep

When to go to bed.

How to use it

Say which end you know. A wake time, or that you are going to bed now.
Read the options. Six cycles down to three.
Prefer more sleep. Total beats hitting a boundary.

When you would use this

Sleep happens in cycles of roughly ninety minutes, and waking between two of them is easier than waking in the middle of one. That is the whole idea, and it is worth being honest about how approximate it is. The arithmetic is simple. Count back from when you have to be up, in blocks of ninety minutes, and add the time it takes to fall asleep. Going to bed and being asleep are different moments, typically ten to twenty minutes apart, and a calculator that ignores that shifts every answer by enough to miss the boundary it was aiming at. Both directions are here, because the question comes both ways: what time should I go to bed, and if I go now, when should I set the alarm. What the ninety minutes is not is a clock. Real cycles run somewhere between seventy and a hundred and twenty minutes, and they lengthen towards morning. So the times below are a reasonable guess at where the boundaries fall and nothing more precise than that. The more important point is that total sleep matters more than the boundary. Six cycles is nine hours and four is six. Waking cleanly on six hours is worse than waking slightly groggy on nine, and the arithmetic here cannot tell which you are doing. Use it to choose between two sensible bedtimes rather than to justify a late one. Persistent trouble sleeping is worth taking to a doctor. This is a calculator and nothing else.

Questions

Are sleep cycles really ninety minutes?: On average, roughly. In practice they run somewhere between 70 and 120 minutes and change through the night, getting longer towards morning. So these times are approximate by construction, and no calculator can tell you where your cycles actually fall on a given night.
Is it better to hit a boundary or get more sleep?: More sleep, nearly always. Six cycles is nine hours and four is six, and waking cleanly on too little is worse than waking a bit groggy on enough. The boundary idea is worth something when you are choosing between two reasonable bedtimes, and not worth cutting an hour for.
Why add time to fall asleep?: Because going to bed and being asleep are not the same moment, and the gap is typically ten to twenty minutes. Leaving it out shifts every answer by that much, which is enough to land you in the middle of a cycle rather than between two.