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### How to Solve Clock Angle Problems Geometrically

Related Topics: Clock Angle Problem Formula

Suppose we are to determine the measure of angle θ in Fig. 1, an analog clock showing the hands at 4:42. One approach to solving for θ is to find the measures of angle A and angle B with respect to the 12 o’clock position as shown in Fig. 2, and then compute the difference between the two angles—the result is the measure of θ.

 Fig. 1. Clock at time 4:42; θ is the angle between the hands. Fig. 2. The difference between  angles A and B is equal to θ.

Finding the measure of angle A

The minute hand of an analog clock starts the hour at the 12 o’clock position and rotates one complete circle in 60 minutes. Dividing 360° by 60 minutes, we obtain 6° per minute—this tells us that the minute hand rotates
every minute, Fig. 3.

 Fig. 3. A shade of color above is 6°. One hatch mark in a basic analog clock  is equivalent to 6°. Fig. 4. The minute hand makes an angle A with respect to the 12 o'clock position.

So, to determine the measure of angle A as shown in Fig. 4, we have to multiply the time past the hour by 6 degrees. If the time is 4:42, then the measure of the angle between the minute hand and the 12 o’clock position is

The formula for solving angle A, in degrees, is:

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

Finding the measure of angle B
On the other hand, the hour hand starts the hour at an hour mark and rotates to the next hour mark in one hour or 60 minutes. Since there are 12 hour marks in an analog clock, the measure between any two adjacent hour marks is
360°12 = 30°, see Fig. 5. Dividing 30° by 60 minutes we obtain 0.5 degrees per minute—this tells us that the hour hand rotates 0.5 degrees every minute.

 Fig. 5. The hour hand rotates 30° in one hour or 60 minutes. Fig. 6. The hour hand rotates an angle x at time M past the hour.

So, at a time M past the hour, the hour hand rotates an angle x from its starting position, Fig. 6. (Angle x is obviously less than 30 degrees unless M is equal to 60 minutes.) The measurement of this angle x, in degrees, is equal to 0.5 times the time past the hour. In our example, the time is 4:42, therefore the measure of angle x is

The formula for solving angle x, in degrees, is:

 2

Now, to determine the measure of angle B in Fig. 7, we have to add angle x and the angle between the 12 o’clock position and the hour hand’s starting position—the measure of the latter angle is equal to 30 times the hour part of the given time or 30
×H, Fig. 8.

 Fig. 7. The hour hand makes an angle B with respect to the 12 o'clock position. Fig. 8. Angle B is equal to the sum of angle 30H and angle x. The value of H above is 4.

Therefore, at 4:42, the measure of angle B is:

The formula for solving angle B, in degrees, is:

 3 where H is the hour part of the given time, in hours.

Solving the measure of angle θ
Finally, we can solve for
θ by subtracting A from B. See Fig. 2.

The formula for solving θ is:

 4 The absolute value ensures a positive result for θ.