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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]

where:

  • θ is the angle in degrees of the hand measured clockwise from the 12
  • MΣ is the minutes past 12 o'clock.
  • H is the hour.
  • M is the minutes past the hour.

Equation for the angle of the minute hand[edit]

where:

  • θ is the angle in degrees of the hand measured clockwise from the 12 o'clock position.
  • M is the minute.

Example[edit]

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[edit]

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

where

  • H is the hour
  • M is the minute

If the angle is greater than 180 degrees then subtract it from 360 degrees

Example 1[edit]

The time is 2:20.

Example 2[edit]

The time is 10:16.

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.

H is an integer in the range 0–11. This gives times of: 0:00, 1:05.45, 2:10.90, 3:16.36, 4:21.81, 5:27.27. 6:32.72, 7:38.18, 8:43.63, 9:49.09, 10:54.54, and 12:00. (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]