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

By BYJU'S Exam Prep

Updated on: September 25th, 2023

The number of times the hands of a clock are at right angle in a day is 44. A right angle is a 90 degrees angle, similar to a corner of a square. An angle is a figure with two rays, both having a common endpoint. The hands of a clock create a right angle 44 times in a 24-hour day. To find out more about how this number is derived, refer to the solution below.

Table of content

1. Hands of a Clock at Right Angle
2. Steps to Calculate Number of Times Hands of a Clock Make Right Angle
3. How many times are the hands of a clock at right angle in a day?

Hands of a Clock at Right Angle

The hands of a clock create a right angle 44 times in a day. The hands of the clock make 12 rounds throughout the 12 hours. As the hour hand is still and moves slower than the minute hands, this would mean that the minute hand makes 11 revolutions, taking the rotating coordinate system into consideration. So, the points at which right angles are created are 22 times.

Thus, in a 24-hour day, 2 X 22 = 44. Thus, there are 44 right angles created by the hands of a clock in a day. A right angle is anangle with a measurement of 90 degrees.

Steps to Calculate Number of Times Hands of a Clock Make Right Angle

A more mathematical approach to calculating the number of times the hands of a clock make right angles is provided below with proper calculations.

The hands of a clock start moving after midnight. The minute hand moves 360 degrees in 60 minutes. Thus, in a minute, the hands move at 6 degrees. Let us consider this as 6x.

Similarly, the hour hand moves 30 degrees in 60 minutes. That will mean that in a minute it will move at 0.5 degree. Let us also consider this as 0.5x.

In both cases, x is the number of minutes after midnight. The first 90 degree angle is made when the minute hand has moved 90 degrees more than the hour hand. This will give us an equation:

6x = 0.5x + 90

5.5x = 90

x = 16 minutes and 4/11 seconds

So, the first right angle is made about 16 minutes past midnight.

Now, we will check when the minute hand has moved 180 degrees ahead of the hour hand and is 90 degrees behind it. So, we get the following equation:

6x = 0.5x + 270

5.5x = 270

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

At about 11 minutes to 1 o’clock.

Now, to check every time such as situation occurs, we will calculate it as:

49 1/11 – 16 4/11 = 32 8/11 minutes

24 hours is 1440 minutes.

Thus,

1440 X 11/ 360

4 X 11 = 44

Hence, every 24 hours, i.e., in a day there are 44 right angles between the hands of a clock.

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