Clock angle problems are a type of mathematical problem which involve finding the angles between the hands of an analog clock. Questions of this nature may appear in text books, tests, examinations or mathematics competitions.
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Clock angle problems relate two different measurements: angles and time.
A general approach to 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.
where:
where:
The time is 5:24. The angle in degrees of the hour hand is:
The angle in degrees of the minute hand is:
The angle between the hands can also be found using the formula:
The time is 2:40.
where
Hour and minute hands are superimposed only when their angle is the same.
is an integer in the range 0–11. This gives times of: 0:00, 1:05.45, 2:10.90, 3:16.36, etc.
David L. Pagni Angles, Time, and Proportion Mathematics Teaching in the Middle School NCTM May 2005, Volume 10, Issue 9 http://my.nctm.org/eresources/article_summary.asp?from=B&uri=MTMS2005-05-436a