# how many times in a day the two hands of a clock are at 90 degree

### Mohammed

Guys, does anyone know the answer?

get how many times in a day the two hands of a clock are at 90 degree from screen.

## geometry

How many times are the hands of a clock at right angle in a day? Initially, I worked this out to be $2$ times every hour. The answer came to $48$. However, in the cases of $3$ o'clock and $9$ o'c...

How many times are the hands of a clock at90 90 degrees. Ask Question Asked 9 years ago

Modified 3 years, 6 months ago

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How many times are the hands of a clock at right angle in a day?

Initially, I worked this out to be

2 2

times every hour. The answer came to

48 48 .

However, in the cases of

3 3 o'clock and 9 9

o'clock, right angles happen only once.

So the answer came out to be

44 44 .

Is the approach correct?

geometrypuzzle Share

edited Jun 8, 2019 at 13:25

CommunityBot 1

asked Dec 14, 2013 at 18:01

rnjai 1,7442 2 gold badges 13 13 silver badges 26 26 bronze badges 2

glassdoor.com/Interview/… –

lab bhattacharjee

Dec 14, 2013 at 18:08

In the case of 3 3 and 9 9 o'clock, its still 2 2 times;at 3:00 3:00 sharp and around 3:30 3:30 , at 9:00 9:00 sharp and around 9:30 9:30 . – K. Rmth

Dec 14, 2013 at 18:10

Not quite, @K.Rmth . At

3:30 3:30

the hours hand already advanced a little towards the

4 4

...Anyway it is twice, once at

3:30 3:30

and the other one at some other hour after

3:30… 3:30… – DonAntonio

Dec 14, 2013 at 18:43

Related: Hands of the clock, Revisited. . –

Han de Bruijn

Oct 27, 2021 at 9:16

Add a comment

## 9 Answers

26

Yes, but a more “mathematical” approach might be this: In a 12 hour period, the minute hand makes 12 revolutions while the hour hand makes one. If you switch to a rotating coordinate system in which the hour hand stands still, then the minute hand makes only 11 revolutions, and so it is at right angles with the hour hand 22 times. In a 24 hour day you get 2×22=44.

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answered Dec 14, 2013 at 18:09

Harald Hanche-Olsen 31.1k2 2 gold badges 55 55 silver badges 79 79 bronze badges

Sorry if this should be obvious, but I don't understand how the minute hand makes only 11 rev when the hour hand is still...? –

space

Sep 1, 2017 at 13:17

4

@Helena That's because we are looking at the minute hand in a coordinate system which itself is rotating, in the same direction as the minute hand, but more slowly. So you have to subtract the one full rotation of the coordinate system from the twelve rotations that the minute hand makes in the usual (non-rotating) coordinate system. –

Harald Hanche-Olsen

Sep 2, 2017 at 11:29

Add a comment 5

More mathematically, it can be done as :

The minute hand moves 360 degrees in 60 minutes. This means that the angle of the minute hand is given by 6t, where t is number of minutes past midnight.

The hour hand moves 30 degrees in 60 minutes. This means that the angle of the hours hand is given by 0.5t.

The hands start together at midnight. The first time they make a 90 degree angle is when the minute hand has moved 90 degrees further than the hour hand, so this is given by the equation:

6t = 0.5t + 90 5.5t = 90

t = 16 4/11 (16 minutes and 4/11 seconds)

In other words about 16 minutes past midnight.

The next time is when the minutes hand has gained another 180 degrees on the hour hand, and is 90 degrees behind it:

6t = 0.5t + 270 5.5t = 270

t = 49 1/11 (49 minutes and 1/11 seconds)

At about 11 minutes to 1 o'clock.

For every 180 degrees that the minute hand gains on the hour hand there will be one 90 degree angle, so every 49 1/11 - 16 4/11 = 32 8/11 minutes

24 hours is 1440 minutes. 1440/(32 8/11 ) = 44

So every 24 hours there are 44 right angles between minute hand and second hand.

Hope it helps :) Share

answered Jun 30, 2016 at 10:26

Ankit Kaushik 511 1 silver badge 2 2 bronze badges Add a comment 5

The question reads, "How many times are the hands of a clock at right angle in a day?"

There are three hands on a clock.

The number of occurrences of right angles between the hour and minute hands in a 24-hour period is 44.

The number of occurrences of right angles between the minute and second hands in a 24-hour period is 2832.

The number of occurrences of right angles between the second and hour hands in a 24-hour period is 2876.

Soooo-ooo: 5752.

To the down-voter(s): Is my answer wrong, or do you not like the typography, or what?

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edited Aug 28, 2016 at 5:04

answered Aug 28, 2016 at 4:17

Senex Ægypti Parvi 2,3691 1 gold badge 13 13 silver badges 16 16 bronze badges

I like this answer and the way you addressed the down voters –

Sathasivam K

Aug 28, 2016 at 5:05

2

Your answer needs a better explanation. As to why is it 2832 and 2876. –

Alex

Jul 28, 2020 at 10:59

Add a comment 2

The following C code prints out the times of the day (12hr cycle) that the hands make 90degree angles (to the nearest 3 seconds to avoid the need for floating point). It is indeed 44 times.

The pattern actually repeats itself every 6 hours.

स्रोत : **math.stackexchange.com**

## [Solved] How many times the hands of a clock are at right angles to r

The minute hand covers 360° in an hour and 6° in a minute. The hour hand covers 30° in an hour and 0.5° in a minute. Calculation: Starting fr

Home Logical Reasoning Clock and Calendar

## Question

Download Solution PDF

## How many times the hands of a clock are at right angles to reach other in 24 hours?

22 24 44 48

## Answer (Detailed Solution Below)

Option 3 : 44

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## Detailed Solution

Download Solution PDF

The minute hand covers 360° in an hour and 6° in a minute.

The hour hand covers 30° in an hour and 0.5° in a minute.

**Calculation:**

Starting from midnight i.e. 12 o clock at the midnight, the first time the difference between the two hands would be 90° is:

6x = 0.5x + 90° (x is the number of minutes)

⇒ 5.5x = 90°

⇒ x = 180/11 minutes

The next time the difference between the two hands would be 90° is when the minute hand would have moved 180° away from the hour hand or the difference between both hands would have been 270°.

6x = 0.5x + 270 ⇒ 5.5x = 270

⇒ x = 540/11 minutes

Thus, the difference between two consecutive moments where both hands forms a right angle is:

540/11 - 180/11 = 360/11 minutes

Thus, the two hands form a right angle after every 360/11 minutes.

Total number of minutes in a day = 24 × 60 = 1440 minutes.

Number of times the two hands will form a right angle in a day = 1440 ÷ (360/11)

⇒ (1440 × 11) ÷ 360 ⇒ 4 × 11 = 44 times

Hence, the two hands will form right angle **"44 times"** in a day.

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## How many times in a day, the two hands of a clock are at 90°?

Answer (1 of 5): On the hour, the two hands of the clock are at 90 deg to each other at 3 am. 9 am 3 pm and 9 pm, that is 4 times a day. However there will be a total of 44 times a day that the two hands are at 90 deg. These occurrences are at 12:16:21.818 am and 12:49 am 1:21:49.09 am and 1:5...

How many times in a day, the two hands of a clock are at 90°?

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Sort HsBadarinath

Ph.D. in Civil Engineering. Maths keeps one mentally active.Author has 57.6K answers and 35.7M answer views3y

On the hour, the two hands of the clock are at 90 deg to each other at 3 am. 9 am 3 pm and 9 pm, that is 4 times a day.

However there will be a total of 44 times a day that the two hands are at 90 deg. These occurrences are at

12:16:21.818 am and 12:49 am

1:21:49.09 am and 1:54:32.73 am

2:27:16.36 am

3:00:00 am and 3:32:43.636 am

4:05:27.27 am and 4:38:10.909 am

5:10:54.54 am and 5:43:38.18 am

6:16:21.818 am and 6:49:5.454 am

7:21:49.09 am and 7:54:32.73 am

8:27:16.36 am

9:00:00 am and 9:32:43.636 am

10:05:27.27 am and 10:38:10.909 am

11:10:54.54 am and 11:43:38.18 am

12:16:21.818 pm and 12:49 pm

1:21:49.09 Related questions

How many times the hour hand and minute hand form 90° in a day?

How many times in a 24-hour period do the hour and minute hands of a clock form a right angle?

How many times in a day do the two hands on a clock coincide?

How many right angles are in 24 hours?

At what time after 2 o'clock will the hour of minute hands be at 90° for the 2nd time?

Warrior

Lives in Ujjain, Madhya Pradesh, India (2000–present)3y

Simply answer is 48 Times.

A detailed answer of how? Is given below.

We will talk about how many times the hour hand and minute hand form a 90° angle with each other in an hour say 12:00 a.m. to 01:00 a.m and then multiply it with 24 (since 24 hours a day) to get our answer of how many times they do so in a day.

Okay! Let's just understand a simple thing first (If you didn't thought about it yourself).

If it's 12:15 a.m. does the hour hand and the minute hands are at 90° with each other? Seems obvious yes but NO. ( If you already realise this skip to the para starting with bold letter Now)

In one m

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Dharmendra Singh

Studied Metallurgy and Material Science (Graduated 2014)Author has 299 answers and 500K answer views3y

The correct answer is 44 times

In each hour minute hand makes 90° angle two times

for example time between 12 to 1 first 90° angle will be at 12:16 (approx) and second time at 12:50 (approx)

But between the time 2 to 4 and 8 to 10 it makes 90° angle only three times

So in 12 hours it makes 90° angle 22 times

In one day or 24 hours it will make 90° angle 44 times

Aman

12 from Government Sarvodaya Co-ed Vidyalaya, B-4, Paschim Vihar, New Delhi3y

If we talk about the two hands which shows minutes and hours then it will make 48 times as twice in one rotation and 24 *2=48 times.

48 TIMES. Related questions

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Bhavini Rohit

Business Development Officer (2014–present)3y

24

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Nathan Wilde Upvoted by Michael Jørgensen

, PhD in mathematics4y

Related

Is there a time when all 3 hands on a clock are 120 degrees apart?

Today I wrote an Excel spreadsheet trying to figure this out, with the restriction that all hands are not continuous and only move each second. Here are my steps:

1. Translated each second in a twelve-hour clock cycle to a location for each hand in degrees.

2. Subtracted each hand's location from the others and then took the absolute value (because sometimes one hand is head of another and sometimes reversed, but we just want distances).

3. Used the "if" function to find values between 119 and 121, and then 239 and 241 (You need the 240 difference because sometimes the subtraction in step 2 will

Harshit Sethi

Works at Qualcomm India (company)5y

Related

How many times a day do the hands of a clock overlap?

Check How many times a day do a clock’s hands overlap?

If that doesn’t explain well, check the below math I did to solve it :

1 day = 24 hours. Minute hand:-

In 60 minutes covers 360 degree, so speed of minute hand is 6 degree per minute.

Hour hand:-

In 12 hours covers 360 degree, so speed of hour hand is 30 degree per hour or 1/2 degree per minute.

Angular relative speed between minute and hour hand = 6–1/2 = 11/2 degree per minute.

Let’s see when do minute hand and hour hand overlap, basically start with a reference of 12:00 AM, At this point both are at same position but will now move at different

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