Angles in a Clock (#1)

This is one of the very interesting topics, which can be used to learn and practice more about angles. Almost all of us wear watches and have clocks on the walls. A very simple circle, divided into 12 parts that give us the correct time, according to which we plan our daily tasks and maintain our natural rhythm.

We all know that the circle has an angle of or radians.

There are generally three hands on the clock. The “hour” hand is the slowest and the smallest of all three. It completes one revolution around the circle every twelve hours. The second hand is the “minute” hand. This is the first division of the clock; hence, the name “minute hand.” The minute hand completes the revolution around the clock once every hour. The third is the second hand. It is the second division of the clock, hence the name for the fastest and longest hand on the clock. It completes its revolution around the clock once every minute.

60 seconds make 1 minute, and 60 minutes make 1 hour.
Therefore 1 hour =

Every hour, therefore, is equal to

Example 1:
$$ \text{When minute hand moves halfway around the circle, i.e. 30 minutes and 180 degrees,} \newline \text{the hour hand moves halfway between two digits.} \newline \text{The hour hand travels 15 degrees from its original position.} \newline \text{Hence, the hour hand moves $\frac{15}{30}$ = 0.5 degrees per minute.}$$

Example 2:
$$ \text{One can verify that when minute hand is on 12 and the hour hand is on 3,} \newline\text{the angle between them is 90$^\circ$.} $$

$$ \text{One can also verify that when minute hand is on 12 and the hour hand is on 6,} \newline\text{the angle between them is 180$^\circ$.} $$

$$ \text{One can also verify that when minute hand is on 12 and the hour hand is on 12,} \newline\text{the angle between them is 0$^\circ$.} $$

Example 3:
From the given information above, one can find out the angle between minute hand and the hour hand, keeping in mind that the hour hand moves proportionally to the movement of the minute hand. Hence, when the minute hand is at a certain position, say ‘x’ minutes, the hour hand has moved ‘x/2’ degrees from the digit position.

3.1 Find the angle between minute hand and hour hand at 3:45
A: At 45 minutes, the minute hand is on 9 and the hour hand is somewhere between 3 and 4.
We will first find the angular distance between 3 and 9. It is six hour divisions, hence 30*6 = 180 degrees.
But, since the hour hand moves in proportion to the minute hand (0.5 deg per minute), the hour hand has moved
So, we will have to subtract the angular distance moved by hour hand from the angular distance between 3 and 9. Therefore the final angle is

3.2 Find the angle between minute hand and hour hand at 10:20
A: At 20 minutes, the minute hand is on 4 and the hour hand is somewhere between 10 and 11.
We will first find the angular distance between 10 and 4. It is six hour divisions, hence 30*6 = 180 degrees.
But, since the hour hand moves in proportion to the minute hand (0.5 deg per minute), the hour hand has moved
So, we will have to subtract the angular distance moved by the hour hand from the angular distance between 10 and 4. Therefore the final angle is

3.3 Find the angle between minute hand and hour hand at 4:38
A: At 38 minutes, the minute hand is between 7 and 8, and the hour hand is somewhere between 4 and 5.
We will first find the angular distance between 4 and 7. It is three hour divisions, hence 30*3 = 90 degrees.
But, since the hour hand moves in proportion to the minute hand (0.5 deg per minute), the hour hand has moved
So, we will have to first add the angular distance covered in those 3 minutes by the minute hand, add it to 90, and then subtract the angular distance moved by the hour hand. Therefore, the final angle is:
$$ \text{Step 1:} \space 90 + 18 = 108^\circ $$
$$ \text{Step 2:} \space 108 – 19 = 89^\circ \space \text{(Final answer)} $$

In this manner, one can understand various angles that the hour hand and minute hand make with each other at a specific time. For understanding and practicing more such interesting topics, consider subscribing to UniverseUnlocks. Keep practicing, keep getting better. All the best!