Clock angle problem

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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.

Math problem[edit]

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 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.[1]

Equation for the angle of the hour hand[edit]

\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
  • \scriptstyle H is the hour.
  • \scriptstyle M is the minutes past the hour.
  • \scriptstyle M_\Sigma is the minutes past 12 o'clock.

Equation for the angle of the minute hand[edit]

\theta_{\text{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[edit]

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[edit]

The angle between the hands can 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}

where

  • \scriptstyle H is the hour
  • \scriptstyle M is the minute

Example[edit]

The time is 2:20.

\begin{align}
\Delta\theta 
 &= \left|\frac{1}{2}(60 \times 2 - 11 \times 20)\right|\\
 &= \left|\frac{1}{2}(120 - 220)\right|\\
 &= 50
\end{align}

When are the hour and minute hands of a clock superimposed?[edit]

The 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. (0.45 minutes are exactly 27.27 seconds.)

See also[edit]

References[edit]

  1. ^ Elgin, Dave (2007). "Angles on the Clock Face". Mathematics in School (The Mathematical Association) 36 (5): 4-5. JSTOR 30216063. 

External links[edit]