What is the clock angle? Let us examine the relationships between the hands on an analog clock and their angles calculated in degrees. I will examine both a 12-hour dial and a 24-hour dial. I will also present a nice interactive animation to help you understand the relationships.

Note: The 12-hour clock will have its "12" or "0" at the top and "6" at the bottom. The 24-hour clock will have its "24" or "0" at the top and "12" at the bottom.

Animation of clock angles

Before we dig into the theory, let us first get a visual overview of what clock angles are. In figure 1 we can see an animation of the angles that each hand represent.

NB! There is an improved implementation of the interactive clock angle animation, namely the NNM Clock angle, which I recommend to use instead of the animation below. Both are working perfectly, but the NNM Clock angle is probably a bit more pedagogic, have more functionality and is easier to interact with.

Note: Toggling the digital clock in the animation, between 24-hour and 12-hour AM/PM, is not an option. That is because the 12-hour clock with AM/PM is not an option.

Note: The 12-hour clock will of course use the numbers 0-11 when Zulu time is between 00:00:00 and 11:59:59 and the numbers 12-23 when Zulu time is between 12:00:00 and 23:59:59.

Definition

Let \(h\), \(m\) and \(s\) denote the current number of hours, minutes and seconds respectively. The hours are, of course, numbered 0-23. The minutes and seconds are numbered 0-59.

Denote the angle for the hour hand \(H^{12}\) for 12-hour dial and \(H^{24}\) for 24-hour dial, the angle for the minute hand \(M\) and the angle for the second hand \(S\). The angles are all measured in degrees and are calculated clockwise from the top.

Note: We will use \(H\), instead of \(H^{12}\) or \(H^{24}\), when speaking about the angle of the hour hand in general terms and when it does not matter if it is a 12-hour or 24-hour dial.

Denote the velocity of the hands per time unit as \(X_y\), where \(X\) is the hand (\(H\), \(M\) or \(S\)) and \(y\) is the time unit (\(h\), \(m\) or \(s\)). The velocity is measured in degrees per time unit. For example, \(M_s\) denotes the number of degrees that the minute hand moves per second and \(H_m\) denotes the number of degrees that the hour hand moves per minute.

The second hand

The second hand rotates one turn, 360°, in 60 seconds. The rotation is therefore \(\frac{360°}{60} = 6°\) per second. We also have one turn per minute or 60 turns per hour. This means that it takes \(\frac{1}{6}\) seconds for the second hand to rotate 1°.

So, for the second hand we have the following three velocities:

The angle for the second hand, \(S\), equals 0° at 0 seconds, 90° at 15 seconds, 180° at 30 seconds and 270° at 45 seconds. Every 30° for the second hand equals 5 seconds.

The minute hand

The minute hand rotates one turn, 360°, in 60 minutes. The rotation is therefore \(\frac{360°}{60} = 6°\) per minute. We also have one turn per hour or \(\frac{1}{3600}\) turns per second. This means that it takes 10 seconds for the minute hand to rotate 1°.

So, for the minute hand we have the following three velocities:

The angle for the minute hand, \(M\), equals 0° at 0 minutes, 90° at 15 minutes, 180° at 30 minutes and 270° at 45 minutes. Every 30° for the minute hand equals 5 minutes.

The hour hand

The hour hand is special and we need to distinguish between the two types of clock faces.

12-hour clock

The hour hand rotates one turn, 360°, in 12 hours. The rotation is therefore \(\frac{360°}{12} = 30°\) per hour. We also have \(\frac{1}{720}\) turns per minute or \(\frac{1}{43200}\) turns per second. This means that it takes 2 minutes (120 seconds) for the hour hand to rotate 1°.

So, for the hour hand, in the 12-hour dial, we have the following three velocities:

The angle for the hour hand, \(H^{12}\), equals 0° at 0 (and 12) hours, 90° at 3 (and 15) hours, 180° at 6 (and 18) hours and 270° at 9 (and 21) hours. Every 30° for the hour hand equals 1 hour.

24-hour clock

The hour hand rotates one turn, 360°, in 24 hours. The rotation is therefore \(\frac{360°}{24} = 15°\) per hour. We also have \(\frac{1}{1440}\) turns per minute or \(\frac{1}{86400}\) turns per second. This means that it takes 4 minutes (240 seconds) for the hour hand to rotate 1°.

So, for the hour hand, in the 24-hour dial, we have the following three velocities:

The angle for the hour hand, \(H^{24}\), equals 0° at 0 hours, 90° at 6 hours, 180° at 12 hours and 270° at 18 hours. Every 15° for the hour hand equals 1 hour.

Note: As expected, the velocity for the hour hand in a 12-hour clock is twice the velocity for the hour hand in a 24-hour clock.

Velocities

Of course, the "three" velocities for each hand are in fact the same. It is the same quantity, denoted with different numerical values and units. We can see that the numerical value for the velocity per hour for each hand is always 60 times higher than the numerical value for the velocity per minute. The velocity per minute numerical value is always 60 times higher than the velocity per second numerical value. Finally, the numerical value for the velocity per hour is always 3600 times higher than the numerical value for the velocity per second. All because one hour equals 60 minutes and one minute equals 60 seconds.

Note: It is the same as 90 km/h denotes the same speed as 25 m/s. Same speed, different numerical values and units. Or that a temperature of 0 degrees Celsius equals 32 degrees Fahrenheit.

These relationships can be expressed as

where \(X\) is one of the hands (\(H\), \(M\) or \(S\)). This is true for both the 12-hour dial and the 24-hour dial.

Smooth or discrete

The hands can rotate either smooth or discrete. When discrete the hands "ticks" and when smooth the hands rotates continuously.

Let \(u\) denote the current number of milliseconds (numbered 0-999).

We have the following formulas for calculating the angles:

Note: When calculating \(H^{12}\), we have to use modulo 12 of the hour value, since we are using 12-hour dial.

References

• Clock angle problem
• 12-hour clock
• 24-hour clock
• 24-hour analog dial
• Modular arithmetic
• MathJax
• LaTeX/Mathematics