Clock angle problem

The diagram shows the angles formed by the hands of an analog clock showing a time of 2:20

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.

Math problem

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.

Equation for the angle of the hour hand

$\theta_{\text{hr}} = \frac{1}{2}M_\Sigma = \frac{1}{2}(60H + M)$

where:

• $\scriptstyle\theta$ is the angle in degrees of the hand measured clockwise from the 12 o'clock position.
• $\scriptstyle H$ is the hours past 12 o'clock.
• $\scriptstyle M$ is the minutes past the hour.
• $\scriptstyle M_\Sigma$ is the minutes past 12 o'clock.

Equation for the degrees on the minute hand

θ_min. = 6M

where:

• $\scriptstyle\theta$ is the angle in degrees of the hand measured clockwise from the 12 o'clock position.
• $\scriptstyle M$ is the minute.

Example

The time is 5:24. The angle in degrees of the hour hand is:

$\theta_{\text{hr}} = \frac{1}{2}(60 \times 5 + 24) = 162$

The angle in degrees of the minute hand is:

$\theta_{\text{min.}} = 6 \times 24 = 144$

Equation for the angle between the hands

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

$\begin{align} \Delta\theta &= \left|\theta_{\text{hr}} - \theta_{\text{min.}}\right| \\ &= \left|\frac{1}{2}(60H + M) - 6M\right|\\ &= \left|\frac{1}{2}(60H - 11M)\right| \end{align}$

Example

The time is 2:40.

$\begin{align} \Delta\theta &= \left|\frac{1}{2}(60 \times 2 - 11 \times 40)\right|\\ &= \left|\frac{1}{2}(120 - 440)\right|\\ &= 160 \end{align}$

where

• $\scriptstyle H$ is the hour
• $\scriptstyle M$ 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.

$\begin{align} \theta_{\text{hr}} &= \theta_{\text{min.}}\\ \Rightarrow \frac{1}{2}(60H + M) &= 6M\\ \Rightarrow 11M &= 60H\\ \Rightarrow M &= \frac{60}{11}H\\ \Rightarrow M &= 5.\overline{45}H \end{align}$

$\scriptstyle H$ is an integer in the range 0–11. This gives times of: 0:00, 1:05.45, 2:10.90, 3:16.36, etc.

See also

• Clock position

Notes and references

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