[Math] Finding the exact time the hands of a clock form a given angle

anglecirclesgeometry

Is there any method of finding what time a clock will form a certain angle? For example, what is the first time after 5:00 that the minute and hour hands of a clock will for a 54° angle? Thanks.

Best Answer

The equation for the angle between the hands:

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[Math] Parametric equation of clock hands

This is NOT a proper answer to the request on parametric equations. However, if only the locations of the tips of the hands are of interest, you can consider the following:-

Let the lengths of the minute and hour hands be $M$ and $H$ respectively.

After 60 minutes = 1 hr, the minute hand moved $2 \pi M(1)$.

After 30 minutes = 0.5 hr, the minute hand moved $2 \pi M(0.5)$

After x minutes; (where $x \le 60$) $= \frac {x}{60}$ hour, the minute hand moved $2 \pi M(\frac {x}{60})$.

After 60 minutes = 1 hour, the hour hand moved $2 \pi H (\frac {1}{12})$.

After y hours (where $y \le 12$) $= \frac {y}{12}$ parts of a 12 hour revolution, the hour hand moved $2 \pi H(\frac {y}{12})$.

[Math] Angle between the hour and minute hands at 6:05

Think of it this way: five minutes after six, the minute hand is $\frac1{12}$ of the circle ahead from 12, while the hour hand has advanced $\frac1{12}$ of the way towards 7 from 6, or $\frac1{144}$ of the circle ahead. The initial angle between the two hands is $\frac12$ of the circle, so the solution is $$\frac12-\frac1{12}+\frac1{144}=\frac{61}{144}=152.5^\circ$$

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Related Solutions

[Math] Parametric equation of clock hands

[Math] Angle between the hour and minute hands at 6:05

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