The aim of this tutorial is to demonstrate the derivation of the formula
for finding clock angles in analog clocks. By “clock angle” we mean the
measurement of angle θ whose region does not include the 12 o'clock position as shown in Fig.1; θ is not necessarily an acute angle. The steps necessary to find the measure of angle θ at time t are:
 Finding the measurement of angle A measured clockwise from the 12 o'clock position, Fig. 2.
 Finding the measurement of angle B measured clockwise from the 12 o'clock position, Fig. 2.
 Calculating the absolute value of the difference of angles A and B—the resulting value is the clock angle θ.
Some facts about the basic analog clock
 The face of a basic analog clock is numbered from 1 to 12; the marks corresponding these numerals are called hour marks.
 The measurement of the angle between any two adjacent hour marks is 30°, Fig. 3.
 A clock is divided into 60 marks which are the minute marks—these includes the hour marks.
The behavior of the minute hand
The minute hand rotates at a constant rate and it completes one full circle or 360° in 60 minutes.
Consider Fig. 4. At certain time M past the hour, the minute hand makes an angle A clockwise from the 12 o'clock position.
The measure of angle A makes up a fraction ^{M}⁄_{60 min}
of the complete circle.
This implies that angle A is proportional to the time past the hour or
Solving for the measure of angle A, in degrees, we have

1
where M is the time past the hour, in minutes.


Notice that if M = 1 minute, A = 6°. This means that the minute hand rotates 6° in one minute, 12° in 2 minutes, 18° in 3 minutes, and so forth and so on until it reaches 360° in 60 minutes (see Fig 5).
The behavior of the hour hand
The hour hand rotates at a constant rate. It starts the hour at an hour
mark and after 60 minutes its position is at the next hour mark. The
angle measure between these two hour marks is 30°, thus, the hour hand rotates 30° in 60 minutes. Take note: 30° only in 60 minutes. See Fig. 6.
Consider Fig. 7. At a certain time M past the hour, the hour hand makes an angle x with its starting position.
The measure of angle x makes up a fraction ^{M}⁄_{60 min}
of the total angle the hour hand makes in 60 minutes.
This implies that angle x is proportional to the time past the hour or
Solving for x, in degrees, we have
Notice that if M = 1 minute, x = 0.5°. This means that the hour hand rotates
0.5° in one minute, 1° in 2 minutes, 1.5° in 3 minutes, and so forth and so on until it reaches the next hour mark in 60 minutes. See Fig 8.
The angle of the hour hand clockwise from the 12 o’clock position
In equation 2, x is not the measurement of angle B. To find angle B,
we have to add to x the number of 30degree angles from
the 12 o'clock position clockwise to the hour hand’s starting position. The
measure of angle B is:
See Fig. 9. The formula for finding the measure of angle B is:

3
where H is the hour part of the time.


The angle θ between the minute and hour hands
The absolute value of the difference of A and B gives us the value of θ:
Replacing A and B with equations 1 and 3, respectively, θ becomes
Thus, the
measurement of angle θ,
in degrees, is

4
The
absolute value symbol ensures a positive result.


We can rewrite θ as
Example: What is the measurement of the angle between the minute and hour hands when the time is 4:42? See Fig. 1.
Solution: Let H = 4 and M = 42, then
Therefore, the measure of the angle between the minute and hour hands at 4:42 is 111°.
Note: If we are to determine angle β (beta, the angle that contains the 12 o'clock position) as shown in Fig. 10, then use the formula 5.

Fig. 1
Fig. 2. Angle A minus
angle B is θ.
Fig. 3. Angle between
adjacent hour marks is 30°.
Fig. 4. Angle A measured
clockwise from the
12 o'clock position.
Fig. 5. The minute hand
rotates 6° every minute.
Click image to enlarge.
Fig. 6. The hour hand rotates 30° in 60 minutes.
Fig. 7. The hour hand rotates x° at time M past the hour.
Fig. 8. Click image to enlarge.
Fig. 9. Angle B is equal
to the sum of 30×H and
angle x.
Fig. 10. Clock showing
