Clock angle problem
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.
Mathematical content
Clock angle problems relate two different measurements - angles and time. To answer the problem the relationship between the time shown (or an elapsed time) and the position of the hands (as given by an angle) has to be found.
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 degrees in 12 hours. This is equivalent to 360 degrees in 720 minutes or 0.5 degrees per minute. The minute hand turns 360 degrees in 60 minutes or 6 degrees per minute.
Equation for the degrees on the hour hand
(0.5 degrees per minute on the hour hand) * (the time on the hour hand * 60 minutes per hour) + (0.5 degrees per minute on the minute hand) * (the time on the minute hand)
Equation for the degrees on the minute hand
(6 degrees per minute on the minute hand) * (the time on the minute hand)
'Example: The time is 5:24'
The degree on the hour hand is (0.5*5*60)+(0.5*24)=162 degrees
The degrees on the minute hand is 6*24=144 degrees
Equation for the degrees between the hands
The angle between the hands can also be found using the formula cos-1(cos(5.5x)), where x=the number of minutes past noon. This will always give an angle between 0 and 180 degrees.
'Example: The time is 1:35'
1:35 is 1(60)+35=95 minutes past noon.
cos-1(cos(5.5*95))=cos-1(cos(522.5))=cos-1(-.95372)=162.5 degrees between the hands
Examples
- "If the clock shows three o’clock, what degrees are the hands showing?"1
- "What is the measure of the angle between the hands on a clock if one hand is on the number 12 and the other is on the number 1?"2
- " Once, excitedly explaining how to solve a word problem about angles on a clock, Mr. Cook yanked the classroom clock off the wall and moved the hands to show us. "At three o'clock, it's a 90-degree angle," he said. "At five, it's 150 degrees between the big hand and the little hand." Mr. Cook's helpful demonstration broke the clock. It never again kept time."3
Notes and references
Footnotes
- NCTM Illuminations "Junior Architect" http://illuminations.nctm.org/Lessons/Architect/Architect-AS-ProbSolvTasks.pdf
- NCTM Figure This http://www.figurethis.org/pdf/ch/challenges_9-12.pdf
- Bonnie Wallace "The Day Mr. Smith Bought Math Into This World" Science Notes Winter 1995 http://scicom.ucsc.edu/scinotes/9502/Geometry.html
General references
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