Do you remember last century back in 4th grade when the teacher tested everyone to find all 4 times the clock’s hour and minute hand are at right angles?
She said 3 and 9 o’clock were right angles but we had to figure out the other two, and she didn’t accept my answer of 6:161. Turns out she was wrong; there’s 20 other times the hands are at right angles! Take that2 Mrs. Ditty! But before getting to that, just what is the amount of time for a minute hand to return to its starting alignment with the hour hand?
For every hour, the minute hand has to make a full turn around the clock. But as the minute moves, so does the hour. So once the minute hand has gone around a full hour, the hour hand has moved on. So the minute moves a bit, the hour moves just a tad further. The minute hand moves ahead a little more, and so the hour. This isn’t some Zeno’s paradox here though, as time does move on!
The exact amount of time the minute hand covers to return to its starting alignment with the hour hand is 12/11ths of an hour – that’s one hour, five minutes and 27.27 seconds. You can see this easily (if slowly) by counting the number of times the hour and minute hand overlap each other during the course of twelve hours. This ends up being eleven times for twelve hours, giving the 12/11 ratio. And also that infinite repeating decimal represented by that overhead horizontal bar.
So for the times of when the hour and minute hand are 90 degrees apart, with the hour hand in front of, or leading, the minute hand, this starts at 3 o’clock. With 3 o’clock accounted for, there are 10 other times where this alignment happens. For 9 o’clock like alignments, where the hands are 270 degrees apart (or 90 degrees apart with the hour hand behind, or lagging, the minute hand), there’s an additional ten times when that happens. Thus a grand total of 22 times when the hour and minute hands are at right angles from each other.
I’ve listed these in the table below. I’ve also included the times when the hands overlap (12 o’clock alignments, 0 degrees apart) and when the hands are opposite (6 o’clock alignments, 180 degrees apart).
Clock Hand Alignment Times
| 0 degrees | 90 degrees | 180 degrees | 270 degrees |
|---|---|---|---|
| 12:00:00.00 | 03:00:00.00 | 06:00:00.00 | 09:00:00.00 |
| 01:05:27.27 | 04:05:27.27 | 07:05:27.27 | 10:05:27.27 |
| 02:10:54.54 | 05:10:54.54 | 08:10:54.54 | 11:10:54.54 |
| 03:16:21.81 | 06:16:21.81 | 09:16:21.81 | 12:16:21.81 |
| 04:21:49.09 | 07:21:49.09 | 10:21:49.09 | 01:21:49.09 |
| 05:27:16.36 | 08:27:16.36 | 11:27:16.36 | 02:27:16.36 |
| 06:32:43.63 | 09:32:43.63 | 12:32:43.63 | 03:32:43.63 |
| 07:38:10.90 | 10:38:10.90 | 01:38:10.90 | 04:38:10.90 |
| 08:43:38.18 | 11:43:38.18 | 02:43:38.18 | 05:43:38.18 |
| 09:49:05.45 | 12:49:05.45 | 03:49:05.45 | 06:49:05.45 |
| 10:54:32.72 | 01:54:32.72 | 04:54:32.72 | 07:54:32.72 |
And here they are in a more visual format (times are rounded):
(The code used to make the icon from this post was repurposed to make these infographs)
1. This was a complete guess. I have no memory of what I said for the other one.
2. Or was it Mrs. Markulics? You know what this detail doesn’t really matter.
Orthallelous 
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