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Two Right Angles Each Hour

Published by CHIU Ho Keung in Offbeat
April 13th, 2009

There are two right angles formed by the hour and minute hands each hour. Do you ever notice that on a clock? three o’ clock and nine o’ clock are greedy to have three.


It should be much easier for you this time to understand how to calculate the time when the two right angles appear each hour on a clock after reading my previous two instructions about “overlapping time” and “straight-line time”.

Because we have known the relation between the speed of the hour hand and that of the minute hand as a ratio of 11/12, we use the same principle to track down the two right angles clipped by the two hands. There are 90 degrees in each of them.

The first right angle is solved by 11/12 = 15/? where “ ? ” is the time at which right angle is seen. The second right angle is reckoned by 11/12 = 45/? where “ ? ” is the time at which another right angle is found. It is not hard to remember 15 and 45 in the formulae. 15 means fifteen minutes that form a right angle and 45 means forty-five minutes that take 15 minutes to 60 minutes (one hour). Nonetheless, using 11/12 = 45/? can find the first right- angled time of some o’ clocks as quoted by the example below.

As the calculating process of straight-line time, we have to figure out the overlapping time before finding the right-angled time because it is always fair to start a race when runners (the two clock hands) are ready to run at the same time.

The overlapping time after five o’ clock is 5/11 multiplied by 60. The hands overlap at 05:27:16.36. (05:00:00 plus 00:27:3/11) Do you still remember how to convert the seconds? The second right-angled time is found by 11/12 = 15/? and the answer is 16.36 minutes. Hence the second right angle after five o’ clock happens at 05:43:38.

How about the first right angle? That has to calculate when the overlapping time it is one hour before five o’ clock like the calculating process of straight-line time. The overlapping time of four o’ clock is 04:21:49. When 11/12 = 45/? is solved, the answer is 49.0909 minutes. Therefore the first right angle after five o’ clock appears at 05:10:54.4 by adding 00:49:5.4 to 04:21:49. It seems to be a little confusing but there must be such logical way to find the first and second right-angled time. That is why I put this article at the end of a series of clock time.

When you get hold of the concepts of velocity, displacement and time taken, you will find the mathematics of a clock interesting. At least, you have understood it more right now. Before obtaining a graduation certificate, I have a mission for you and you have to tell me when the hands clip two twenty-degree angles after six o’ clock by writing comments to me.

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