What's the angle between the hour and minute hand at 3:27?

Question

 What's the angle between the hour and minute hand at 3:27?

Answer 1

Let’s determine the angle in degrees from midnight=0 degrees going clockwise for each of the hands.

The time 3:27 is 3 hours and 27 minutes. For the minute hand, this is 27 minutes from the on-the-hour angle of zero degrees, so the angle is (27 minutes)*((360 degrees)/(60 minutes)) = 162 degrees. [Note: 30 minutes is 180 degrees, so this makes sense.]

For the hour hand, (3 hours)*((360 degrees)/(12 hours)) = 90 degrees plus
(27 minutes)*((360 degrees)/(12 hours))*((1 hour)/(60 minutes)) = 13.5 degrees. So, a total of 103.5 degrees.

You could also calculate the position of the hour hand by:
(207 minutes)*((360 degrees)/(12 hours))*((1 hour)/(60 minutes)) = 103.5 degrees. [Note: 3 o’clock is 90 degrees, so this makes sense.]

The difference: 162 - 103.5 = 58.5 degrees.

Answer 2

The hour hand moves (5/12) of a circle every hour.    So at 3:27, the hour angle is at (27/60)*(30 degrees/hr)  degrees past the number 3.   13.5 degrees

 

On the other hand, the minute hand is at (12 min) *  (6 degrees/min)  past the number 3 at the same instant.   72 degrees.

 

The angle between the minute hand and the hour hand is 72-13.5 = 58.5 degrees.