The Angle of Time

Ever wondered what the time would be when clock hands have a certain angle between them?  Yes me too! All the time! After pondering the issue I posed it to my father, who came up with an elegant formula.

Given an hour (H) between 0 and 11 and and angle (A) between +/-360°, the minutes (M) before or past before the hour is given by:

M = (60 x H /11) + (12A/66)

Give it a try for some different input values here.  You’ll see some interesting anomalies around 11 and 12 o’clock.  Try the values 11, 10 and “after”.  You might think the answer ought to be a few minutes to 11. But it’s actually just after 12. That’s because I defined “after” as after the minute hand passes the hour hand, and the latter doesn’t catch up to the former until 00:00.

 Hour: Angle:
Before After

'Before' means the angle between the minute to hour hand before the minute hand gets to the hour hand. 'After' is the angle between the hour and minute hand after the minute hand passes the hour hand, starting from the given hour.

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