Clock Angle Help

 

The following strategy for finding the angle measure between the hands on an analog clock has helped two or three students who have come into my office for individual help. Possibly this will help you.

 

 

Let’s consider the time 6:17. On the clock face, the hour hand is between the 6 and 7, and the minute hand is between 3 and 4. Here’s the strategy:

 

  1. Mark the full divisions that contain the angle between the hands on the clock. You should mark 7 and 3.
  2. There are four (4) full divisions or 120º because 4(30º) is 4 divisions. So, we know our angle is less than 120º.
  3. Calculate the movement of the hour hand.  (1/2 ) º x 17 min = 8.5º the hour hand has moved from the 6 toward the 7.
  4. HOWEVER, we want to subtract away the angle between the hour hand and the mark on 7 because this angle is NOT between the

hands on the clock.  So, (30 – 8.5) º = 21.5º from the hour hand to the 7 mark.

  1. Now, subtract this from 120º:  (120 – 21.5) º = 98.5º.
  2. Calculate the movement of the minute hand. Since the time is 17 minutes after the hour, the minute hand has moved 2 minutes past 3 toward 4 on the clock. We know that the minute hand moves 6º each minute while the hour hand moves (1/2) º.
  3. We are down to 98.5º. So, in 2 minutes the minute hand moves 6º(2) = 12º away from the mark on 3.
  4. Therefore, (98.5 – 12) º = 86.5º, or the angle between the clock hands at 6:17.

 

On some times for the clock, the number of full divisions will give you an angle that is greater than 180º. This size angle, however, is not a problem because as you subtract away, the angle always becomes < 180º, as we agreed in class that each angle should.