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Geometry 1.4 (extra video) - Angles of hands of a clock - YouTube
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Clock angle problems are a type of mathematical problem which involve finding the angles between the hands of an analog clock.


Video Clock angle problem



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 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

? hr = 0.5 ? × M ? = 0.5 ? × ( 60 × H + M ) {\displaystyle \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
  • H is the hour.
  • M is the minutes past the hour.
  • M? is the number of minutes since 12 o'clock. M ? = ( 60 × H + M ) {\displaystyle M_{\Sigma }=(60\times H+M)}

Equation for the angle of the minute hand

? min. = 6 ? × M {\displaystyle \theta _{\text{min.}}=6^{\circ }\times M}

where:

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

Example

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

? hr = 0.5 ? × ( 60 × 5 + 24 ) = 162 ? {\displaystyle \theta _{\text{hr}}=0.5^{\circ }\times (60\times 5+24)=162^{\circ }}

The angle in degrees of the minute hand is:

? min. = 6 ? × 24 = 144 ? {\displaystyle \theta _{\text{min.}}=6^{\circ }\times 24=144^{\circ }}

Equation for the angle between the hands

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

? ? = | ? hr - ? min. | = | 0.5 ? × ( 60 × H + M ) - 6 ? × M | = | 0.5 ? × ( 60 × H + M ) - 0.5 ? × 12 × M | = | 0.5 ? × ( 60 × H - 11 × M ) | {\displaystyle {\begin{aligned}\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{aligned}}}

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

The time is 2:20.

? ? = | 0.5 ? × ( 60 × 2 - 11 × 20 ) | = | 0.5 ? × ( 120 - 220 ) | = 50 ? {\displaystyle {\begin{aligned}\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{aligned}}}

Example 2

The time is 10:16.

? ? = | 0.5 ? × ( 60 × 10 - 11 × 16 ) | = | 0.5 ? × ( 600 - 176 ) | = 212 ?     ( > 180 ? ) = 360 ? - 212 ? = 148 ? {\displaystyle {\begin{aligned}\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{aligned}}}

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.

? min = ? hr => 6 ? × M = 0.5 ? × ( 60 × H + M ) => 12 × M = 60 × H + M => 11 × M = 60 × H => M = 60 11 × H => M = 5. 45 ¯ × H {\displaystyle {\begin{aligned}\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{aligned}}}

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.)


Maps Clock angle problem



See also

  • Clock position

Clock related angle problem with solutions shortcuts formula ...
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References


Mathcamp321: Geometry - Clock Problems #1 - YouTube
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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

Source of the article : Wikipedia

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