Clock angle problem

${\displaystyle \theta _{\text{hr}}=0.5^{\circ }\times M_{\Sigma }=0.5^{\circ }\times (60\times H+M)}$

where:

• θ 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_Σ is the number of minutes since 12 o'clock. ${\displaystyle M_{\Sigma }=(60\times H+M)}$

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

where:

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

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

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

The angle in degrees of the minute hand is:

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

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

${\displaystyle {\begin{aligned}\Delta \theta &=\vert \theta _{\text{hr}}-\theta _{\text{min.}}\vert \\&=\vert 0.5^{\circ }\times (60\times H+M)-6^{\circ }\times M\vert \\&=\vert 0.5^{\circ }\times (60\times H+M)-0.5^{\circ }\times 12\times M\vert \\&=\vert 0.5^{\circ }\times (60\times H-11\times M)\vert \\\end{aligned}}}$

where

• H is the hour
• M is the minute

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

The time is 2:20.

${\displaystyle {\begin{aligned}\Delta \theta &=\vert 0.5^{\circ }\times (60\times 2-11\times 20)\vert \\&=\vert 0.5^{\circ }\times (120-220)\vert \\&=50^{\circ }\end{aligned}}}$

The time is 10:16.

${\displaystyle {\begin{aligned}\Delta \theta &=\vert 0.5^{\circ }\times (60\times 10-11\times 16)\vert \\&=\vert 0.5^{\circ }\times (600-176)\vert \\&=212^{\circ }\ \ (>180^{\circ })\\&=360^{\circ }-212^{\circ }\\&=148^{\circ }\end{aligned}}}$

In this graphical solution, T denotes time in hours; P, hands' positions; and θ, hands' angles in degrees. The red (thick solid) line denotes the hour hand; the blue (thin solid) lines denote the minute hand. Their intersections (red squares) are when they align. Additionally, orange circles (dash-dot line) are when hands are in opposition, and pink triangles (dashed line) are when they are perpendicular. In the SVG file, hover over the graph to show positions of the hands on a clock face.

The hour and minute hands are superimposed only when their angle is the same.

${\displaystyle {\begin{aligned}\theta _{\text{min}}&=\theta _{\text{hr}}\\\Rightarrow 6^{\circ }\times M&=0.5^{\circ }\times (60\times H+M)\\\Rightarrow 12\times M&=60\times H+M\\\Rightarrow 11\times M&=60\times H\\\Rightarrow M&={\frac {60}{11}}\times H\\\Rightarrow M&=5.{\overline {45}}\times H\end{aligned}}}$

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.)