Clock angle problems are a type of mathematical problem

Mathematical problem

A mathematical problem is a problem that is amenable to being 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 abstract nature, such as Hilbert's...

 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.

Math problem

Clock angle problems relate two different measurements: angles

Angle

In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

 and time

Time

Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....

.

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.

Equation for the angle of the hour hand

where:

• is the angle in degrees of the hand measured clockwise from the 12 o'clock position.
• is the hours past 12 o'clock.
• is the minutes past the hour.
• is the minutes past 12 o'clock.

Equation for the degrees on the minute hand

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:



The angle in degrees of the minute hand is:

Equation for the angle between the hands

The angle between the hands can also be found using the formula:

Example

The time is 2:40.



where

• is the hour
• is the minute

When are hour and minute hands of a clock superimposed?

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., 2:10., 3:16., etc.

Footnotes

1. NCTM Illuminations "Junior Architect" http://illuminations.nctm.org/Lessons/Architect/Architect-AS-ProbSolvTasks.pdf
2. NCTM Figure This http://www.figurethis.org/pdf/ch/challenges_9-12.pdf
3. Bonnie Wallace "The Day Mr. Smith Brought 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

External links

• http://www.delphiforfun.org/Programs/clock_angle.htm
• http://www.eaa.unsw.edu.au/pdf/M_clock-hand-angles.xls
• http://mathforum.org/library/view/7952.html
• http://www.ldlewis.com/hospital_clock/ - extensive clock angle analysis
• http://www.jimloy.com/puzz/clock1.htm