Find the angle between the hour-hand and the minute-hand in circular measure at half past 4?

At 4:30

The hour hand hand has moved 4.5/12*360° or 135°
the minute hand has moved 180°

so the angle between them is 180-135 or 45°

To find the angle between the hour hand and the minute hand at half past 4, we need to determine the positions of the hour and minute hands on the clock.

First, let's determine the position of the hour hand. At half past 4, the hour hand will be pointing exactly at the number 4 on the clock. Since there are 12 hours on a clock, we can represent the position of the hour hand as 4/12 or 1/3 of a full revolution around the clock.

Next, let's determine the position of the minute hand. At half past the hour, the minute hand will be pointing directly at the number 6 on the clock. Since there are 60 minutes in an hour and half past the hour corresponds to 30 minutes, we can represent the position of the minute hand as 30/60 or 1/2 of a full revolution around the clock.

To find the angle between the two hands, we need to calculate the difference between their positions. Since both positions are given in terms of a full revolution around the clock, we can subtract the position of the hour hand from the position of the minute hand:

Angle = Position of minute hand - Position of hour hand
= 1/2 - 1/3
= 3/6 - 2/6
= 1/6

Therefore, the angle between the hour hand and the minute hand at half past 4 is 1/6 of a full revolution around the clock.