The angle between the Hour and Minute hands of the clock (Formula proof)
The hour hand completes two 360° rotations of the clock in a day and the minute hand completes one rotation in 1 hour while the Second hand completes one rotation in 1 minute.
1 Day= 24 Hour
1 Hour= 60 Minutes
1 Minute= 60 Seconds
So, in a day there are 24×60×60=86,400 Seconds.
The circular dial of a clock is divided into 60 equal parts, called Minutes.
60 Minutes ⇒360°
1 Minute ⇒6°
In 1 hour, the hour hand covers 5 such parts which means the rotation of 5×6°=30°
So, the speed of the Hour hand= 30°/60= 0.5 Degrees/Minute
In 1 hour, the Minute hand covers 60 such parts which mean the rotation of 60×6°= 360°
So, the speed of the Minute hand= 360°/60= 6 Degrees/Minute
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When we calculate the angle between the hour hand and the minute hand of a clock, two situations are possible
1. The minute hand is behind the hour hand
2. The minute hand is ahead of the hour hand
In the given figure, the angle between the baseline (0°) and the Minute hand is x° while the angle between the baseline and the Hour hand is y°.
Que: Find the angle between the hour hand and minute hand at H:M am or pm.
*(Here, H means hour and M is minute)
The rotation of Hour hand (x°)= Rotation in H hours + Rotation in M minutes
=Rotation in ‘60×H + M’ Minutes
=(60×H+M)×Speed
=(60×H+M)× 0.5 Degree/Minute
x=30×H+ 0.5×M ………..(i)
The rotation of Minute hand (y°)= Rotation in M minutes
=M×Speed
=M× 6 Degree/Minute
y=6×M ……….(ii)
Here in case:1 the angle between Hands (θ)= y-x
θ=6×M — (30×H+ 0.5×M)
θ=6×M — 30×H- 0.5×M
θ=5.5×M — 30×H
θ=11/2×M — 30×H
In case 2: the angle between Hands (θ)= x-y
θ=(30×H+ 0.5×M)- 6M
θ=30×H- 5.5×M
θ=30×H- 11/2×M
So, the general formula of angle, θ =|30H-11/2M|
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