    # At what time between 3 and 4 oclock is the angle between the minute hand and the hour hand of a clock is 75 degree?

Angle between hour and minute hand at 3:30

Here we will show you how to calculate the angle in degrees made by the hour hand and minute hand on a 12-hour clock at 3:30. Below is an image of what the clock looks like at three-thirty (3:30) so that you can see where the hour hand and the minute hand are located. Before we continue, note that a clock is shaped like a circle and is composed of 360 degrees. There are 60 minutes in an hour, and 360 degrees divided by 60 minutes is 6. Therefore, the minute hand moves 6 degrees per minute. It takes 720 minutes for the hour hand to move around the clock. 360 degrees divided by 720 minutes is 0.5. Therefore, the hour hand moves 0.5 degrees per minute. As you can see, the hour hand and minute hand showing 3:30 on the clock make an angle. We used the hands to make the pie chart below so it is easier to visualize what we are calculating. The hour hand and minute hand on a clock showing 3:30 actually make two angles: The hour hand to the minute hand is outlined in blue, and the minute hand to the hour hand is outlined in red.
Angle from hour hand to minute hand at 3:30 At 3:30, the hour hand has moved 210 out of 720 possible times from the top of the clock. 210 times 0.5 degrees is 105 degrees. At 3:30, the minute hand has moved 30 out of 60 possible times from the top of the clock. 30 times 6 degrees is 180 degrees.

180 - 105 = 75 degrees

Angle from minute hand to hour hand at 3:30

The angle from the minute hand to the hour hand is simply 360 degrees minus the degrees from the hour hand to minute hand that we calculated above.

360 - 75 = 285 degrees.

Now you know how to calculate the degrees of the two angles created by the hour hand and minute hand on a 12-hour clock at 3:30. Again, the two angles created by the hour hand and minute hand at three-thirty are 75 degrees and 285 degrees.
Clock Angle Calculator The angle made by the hour hand and minute hand on a clock at 3:30 is not all we have calculated. We have calculated the clock angles for every minute of the 12-hour clock. Get the degrees for another time here!

Angle between hour and minute hand at 3:31

Here is the next time on our clock that we have calculated the angles for. Check it out! Copyright  |   Privacy Policy  |   Disclaimer  |   Contact

Example 1: What is the smaller angle between the two hands of the clock at 5:00 O’clock?

Sol : At 5 O’clock, the hour hand will be at 5 and the minute hand will be at 12. Now, they have a gap of 5 hour spaces and each hour space is 30 degrees. That means the angle between the two hands will be 30 × 5 = 150 degrees. Example 2: What is the smaller angle between the two hands of the clock at 2:20 pm?

Sol: At 2 O’clock the hour hand will be at 2 and the minute hand will be at 12. Now, they have a gap of 2 hour spaces and each hour space is 30 degrees. That means at 2:00 clock the angle will be = 30 × 2 = 60 degrees. After that in 20 min, the relative coverage of the minute hand will be 20 × 11/2 = 110 degrees. Now take the difference between the two angles, which will become our answer i.e. 110 - 60 = 50 degrees.
Besides that, there is another method. As discussed, in 12 hrs, hour hand travels 360 degrees. Thus, in 2 hrs 20 min i.e. 2 1/3 hrs it will travel (360/12) × (7/3) = 70 degrees. Angle traced by min hand in 60 min = 360 degrees. Angle traced by it in 20 min = 20 × 360/60 = 120 degrees.

Required angle is the difference between the two = 120 – 70 = 50 degrees.

Example 3: At what time between 3 and 4 are the minute and hour hand together?

Sol: At 3 O’clock, the relative distance between the hour and the minute hand is 15 minutes. To catch up with the hour hand, the minute hand has to cover a relative distance of 15 minutes at the relative speed of 11/12 minutes per minute.

Thus, time required = 15/(11/12) = 15 × 12/11 = 180/11 = 16 4/11 min.

Example 4: The minute hand of a clock overtakes the hour hand at intervals of 63 minutes of correct time. How much does the clock lose or gain in a day?

Sol: In a correct clock, the minute hand and hour hand should meet after every 65 5/11 min. But we know that they are meeting after every 63 minutes. So gain in 63 minutes is 27/11 minutes.

Gain in 24 hours =(24 ×60/63) × (27/11) = 56 8/77 min.

Example 5: At what time between 4 and 5 are the minute and hour hand at right angles?

Sol: At 4 O’clock, the relative distance between the hour and the minute hand is 20 minutes. To make a 90–degree angle with the hour hand, the minute hand has to cover a relative distance of 5 minutes at the relative speed of (11/12) minutes per minute. Thus, time required = 5/(11/12) = 60/11 or 5 (5/11) minutes. As explained above in important points, there is still one more case. When a relative distance of 35 minutes has been covered, even then the angles would be at right angles. Time required = 35/(11/12) = 420/11 or 38(2/11) minutes

You can also think that the difference between the two right angles themselves be equal to 30 min. In one of the right angles, the minute hand will be 15 min before the hour hand and in the other; it will be 15 min ahead of the hour hand.

Example 6: A watch gains uniformly. It was observed that it was 6 min slow at 12 o’clock in the night on Monday. On Friday at 6 pm it was 4 min 48 second fast. When was it correct?

Sol:The time between 12 O’clock on Monday night and 6 pm on Friday = 90 hours. Now, the watch gains 6 + 4 4/5 min 6 + 24/5 = 54/5 minutes in 90 hours. So, the watch gains 6 minutes in (90×5×6)/54 = 50 hours

Add 50 hours in Monday 12’O clock night, thus the watch is correct at 2:00 am on Thursday.

Example 7: A clock is set right at 10 am. The clock gains 5 minutes in 12 hours. What will be the true time when the clock indicates 3 pm on the next day?

Sol: Time from 10 am to 3 pm on the following day is 29 hrs. Now, 12 hrs 5 min i.e. 145/12 hrs of this clock = 12 hours of correct clock. So, 29 hours of this clock is (29×12×12)/145 = 144/5 = 28 hours 48 minutes.

So, the time is 12 minutes before 3 pm.

Example 8: The minute hand of a clock overtakes the hour hand at intervals of 65 minutes of correct time. How much does the clock lose or gain in 12 hours?

Sol: In a correct clock, the minute hand and hour hand should meet after every 65 5/11 min. But we know that they are meeting after every 65 minutes. So, the gain in 65 minutes is 5/11 minutes.

Gain in 12 hours = (120×60/65) × (5/11) = 720/143 = 5 5/143 min. Example 9: At what time between 4 and 5, the minute hand will be 2 minutes spaces ahead of hour hand?

Sol: At 4 O’ clock, the two hands are 20 min spaces apart. In this case, the min hand will have to gain (20 + 2) i.e. 22 – minute spaces. So, 22 – minute spaces will be gained in (60/55) × 22 = 24 min.

This activity is about Analog clocks and the angles made by the hands of the clock. You can find out more about angles and how they're measured on the page Degrees (Angles).

### What is the angle between the hands of a clock at 1 o'clock? At 1 o'clock the minute hand (red) points to the 12 and the hour hand (blue) points to the 1. So we need to find the angle between the 12 and the 1.

### How many of this angle are there in a complete turn? There are 12 of them in a complete turn (360°), so each one must be 360° ÷ 12 = 30°

So the angle between the hands of a clock at 1 o'clock is 30° .

Note:

• It doesn't matter whether we are talking about 1 am or 1 pm, the answer is exactly the same for both.
• The angle between the hands at 1 o'clock could also be given as the reflex angle 330°, but we will always give the smaller (acute or obtuse) angle.

### What is the angle between the hands of a clock at 2:30? At 2:30 the minute hand (red) points to the 6 and the hour hand (blue) points halfway between the 2 and the 3. • The angle between the 5 and the 6 is 30°
• The angle between the 4 and the 5 is 30°
• The angle between the 3 and the 4 is 30°
• The remaining angle is ½ × 30° = 15°

So the angle between the hands of a clock at 2:30 = 30° + 30° + 30° + 15° = 105°

### Complete the following table (give the smaller angle in each case):

 Time 1:00 2:30 7:00 10:30 11:20 3:40 5:15 8:45 Angle 30° 105°

## More Complicated Times

Finding the angle between the hands of a clock is easy as long as we don't use complicated times.

For example finding the angle between the hands at 9:37 is much more difficult. You can try that one if you wish, but it's probably too hard.

### Example: At what times is the angle between the hands of a clock equal to 30°?

Notice that the question asks for 'times'. There are many possible answers. Some of them are easy to find, others much more difficult.  1 o'clock 11 o'clock But what about this one? 4:15 PM

At first glance, it looks like this might be a 30° angle also, but at 4:15, the hour hand has already moved a quarter of the way between the 4 and the 5.

So the angle is 30° + ¼ × 30° = 30° + 7½° = 37½° .

This might be a more accurate answer: Can you find more 30° angles like this one?

## At what times of the day do the hands of a clock lie in a straight line?

In other words the angle between them is 180°?

One obvious answer is 6 o'clock: ### But what other answers could there be?

9:15 is not correct for a similar reason that 4:15 didn't give us exactly a 30° angle ... the hour hand has moved on beyond the 9.

This seems to be a very difficult question to answer, but there is an easy way.

### How many times between 6:00 am and 6:00 pm do the hands make a straight line?

There must be at least one point time for each hour:

• one between 7:00 am and 8:00 am,
• one between 8:00 am and 9:00 am,
• one between 9:00 am and 10:00 am,
• etc, up to ...
• one between 4:00 pm and 5:00 pm

That makes 11 equal parts, and so:

12 Hours / 11

= 1 + 1/11 hours

= 1 hour + 60/11 minutes

= 1 hour 5 5/11 minutes

= 1 hour 5 ½ minutes (approximately)

The next time after 6:00 am that the hands make a straight line is about 7:05½ am : Hours 7am - 8am 8am - 9am 9am - 10am 10am - 11am 11am - 12pm 12pm - 1pm 1pm - 2pm 2pm - 3pm 3pm - 4pm 4pm - 5pm Time 7:05½ am

You might like to check your answers on the page Analog and Digital Clocks Animation

## Can you work out the times of the day when the hands of a clock make a right angle?

(Hint: there are 22 of them)

### Answer to the earlier exercise:

 Time 1:00 2:30 7:00 10:30 11:20 3:40 5:15 8:45 Angle 30° 105° 150° 135° 140° 130° 67½° 7½° 