Clock angle problem
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Clock angle problems are a type of
mathematical problem A mathematical problem is a problem that can be represented, analyzed, and possibly solved, with the methods of mathematics. This can be a real-world problem, such as computing the orbits of the planets in the solar system, or a problem of a more ...
which involve finding the angle between the hands of an analog clock.


Math problem

Clock angle problems relate two different measurements:
angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...
s and
time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to ...
. The angle is typically measured in degrees from the mark of number 12 clockwise. The time is usually based on a
12-hour clock The 12-hour clock is a time convention in which the 24 hours of the day are divided into two periods: a.m. (from Latin , translating to "before midday") and p.m. (from Latin , translating to "after midday"). For different opinions on represent ...
. A method to solve such problems is to consider the rate of change of the angle in degrees per minute. The hour hand of a normal 12-hour analogue clock turns 360° in 12 hours (720 minutes) or 0.5° per minute. The minute hand rotates through 360° in 60 minutes or 6° per minute.


Equation for the angle of the hour hand

:\theta_ = 0.5^ \times M_ = 0.5^ \times (60 \times H + M) where: * is the angle in degrees of the hand measured clockwise from the 12 * is the hour. * is the minutes past the hour. * is the number of minutes since 12 o'clock. M_ = (60 \times H + M)


Equation for the angle of the minute hand

:\theta_ = 6^ \times M where: * is the angle in degrees of the hand measured clockwise from the 12 o'clock position. * is the minute.


Example

The time is 5:24. The angle in degrees of the hour hand is: :\theta_ = 0.5^ \times (60 \times 5 + 24) = 162^ The angle in degrees of the minute hand is: :\theta_ = 6^ \times 24 = 144^


Equation for the angle between the hands

The angle between the hands can be found using the following formula: :\begin \Delta\theta &= \vert \theta_ - \theta_ \vert \\ &= \vert 0.5^\times(60\times H+M) -6^\times M \vert \\ &= \vert 0.5^\times(60\times H+M) -0.5^\times 12 \times M \vert \\ &= \vert 0.5^\times(60\times H -11 \times M) \vert \\ \end where * is the hour * is the minute If the angle is greater than 180 degrees then subtract it from 360 degrees.


Example 1

The time is 2:20. :\begin \Delta\theta &= \vert 0.5^ \times (60 \times 2 - 11 \times 20) \vert \\ &= \vert 0.5^ \times (120 - 220) \vert \\ &= 50^ \end


Example 2

The time is 10:16. :\begin \Delta\theta &= \vert 0.5^ \times (60 \times 10 - 11 \times 16) \vert \\ &= \vert 0.5^ \times (600 - 176) \vert \\ &= 212^ \ \ ( > 180^)\\ &= 360^ - 212^ \\ &= 148^ \end


When are the hour and minute hands of a clock superimposed?

The hour and minute hands are superimposed only when their angle is the same. :\begin \theta_ &= \theta_\\ \Rightarrow 6^ \times M &= 0.5^ \times (60 \times H + M) \\ \Rightarrow 12 \times M &= 60 \times H + M \\ \Rightarrow 11 \times M &= 60 \times H\\ \Rightarrow M &= \frac \times H\\ \Rightarrow M &= 5.\overline \times H \end is an integer in the range 0–11. This gives times of: 0:00, 1:05., 2:10., 3:16., 4:21., 5:27.. 6:32., 7:38., 8:43., 9:49., 10:54., and 12:00. (0. minutes are exactly 27. seconds.)


See also

* Clock position


References

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External links

* https://web.archive.org/web/20100615083701/http://delphiforfun.org/Programs/clock_angle.htm * http://www.ldlewis.com/hospital_clock/ - extensive clock angle analysis * https://web.archive.org/web/20100608044951/http://www.jimloy.com/puzz/clock1.htm Mathematics education Elementary mathematics Elementary geometry Mathematical problems Clocks