Clock angle problem

Clock angle problems are a type of mathematical problem which involve finding the angle between the hands of an analog clock.

Clock angle problems are a type of mathematical problem which involve finding the angle between the hands of an analog clock.

Math problem

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 a 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.

Equation for the angle of the hour hand

\theta_{\text{hr}} = 0.5^{\circ} \times M_{\Sigma} = 0.5^{\circ} \times (60 \times H + M) where: is the angle in degrees of the hand measured clockwise from the 12 is the hour. is the minutes past the hour. is the number of minutes since 12 o'clock. M_{\Sigma} = (60 \times H + M)

Equation for the angle of the minute hand

\theta_{\text{min.}} = 6^{\circ} \times M where: is the angle in degrees of the hand measured clockwise from the 12 o'clock position. is the minute.

Example

The time is 2:20. The angle in degrees of the hour hand is: \theta_{\text{hr}} = 0.5^{\circ} \times (60 \times 2 + 20) = 70^{\circ} The angle in degrees of the minute hand is: \theta_{\text{min.}} = 6^{\circ} \times 20 = 120^{\circ}

Equation for the angle between the hands

The angle between the hands can be found using the following formula: \begin{align} \Delta\theta &= \vert \theta_{\text{hr}} - \theta_{\text{min.}} \vert \\ &= \vert 0.5^{\circ}\times(60\times H+M) -6^{\circ}\times M \vert \\ &= \vert 0.5^{\circ}\times(60\times H+M) -0.5^{\circ}\times 12 \times M \vert \\ &= \vert 0.5^{\circ}\times(60\times H -11 \times M) \vert \\ \end{align} where is the hour is the minute If the angle is greater than 180 degrees then subtract it from 360 degrees.

Example 1

The time is 2:20. \begin{align} \Delta\theta &= \vert 0.5^{\circ} \times (60 \times 2 - 11 \times 20) \vert \\ &= \vert 0.5^{\circ} \times (120 - 220) \vert \\ &= 50^{\circ} \end{align}

Example 2

The time is 10:16. \begin{align} \Delta\theta &= \vert 0.5^{\circ} \times (60 \times 10 - 11 \times 16) \vert \\ &= \vert 0.5^{\circ} \times (600 - 176) \vert \\ &= 212^{\circ} \ \ ( > 180^{\circ})\\ &= 360^{\circ} - 212^{\circ} \\ &= 148^{\circ} \end{align}

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

The hour and minute hands are superimposed only when their angle is the same. \begin{align} \theta_{\text{min}} &= \theta_{\text{hr}}\\ \Rightarrow 6^{\circ} \times M &= 0.5^{\circ} \times (60 \times H + M) \\ \Rightarrow 12 \times M &= 60 \times H + M \\ \Rightarrow 11 \times M &= 60 \times H\\ \Rightarrow M &= \frac{60}{11} \times H\\ \Rightarrow M &= 5.\overline{45} \times H \end{align} is an integer in the range 0–11. This gives times of: 0:00, 1:05. , 2:10. , 3:16. , 4:21. , 5:27. . 6:32. , 7:38. , 8:43. , 9:49. , 10:54. , and 12:00. (0. minutes are exactly 27. seconds.)

See also

Clock position

References

External links

https://web.archive.org/web/20100615083701/http://delphiforfun.org/Programs/clock_angle.htm http://www.ldlewis.com/hospital_clock/ - extensive clock angle analysis https://web.archive.org/web/20100608044951/http://www.jimloy.com/puzz/clock1.htm

Text derived from https://en.wikipedia.org/w/index.php?oldid=1369663488 under CC BY-SA 4.0; modifications have been made.