Clock angle problem

Clock angle problems are a type of mathematical problem which involve finding the angle between the hands of an analog clock.

Math problem

Clock angle problems relate two different measurements: angles and time. The angle is typically measured in degrees from the mark of number 12 clockwise. The time is usually based on a 12-hour clock.

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