What is the angle between the hands of a clock at 2:36?
A clock is a circle.
For the big/minute hand to go from 0 to 60 minutes, the hand would have
to go all the way around which would be a rotation of 360 degrees; this
would be the same number of degrees for the little hand after moving
for 12 hours, for example, from 12AM to 12PM. Suppose that at 12
o'clock, there are 0 degrees between the two hands. The degrees between
each hour are 360/12=30 since there are 12 hours. We have 60 minutes in
an hour and so, the degrees between each minute are given by 360/60=6.
Supposing we draw a line from the centre of the clock to hour 12 and
another line from the centre to hour 2, the degree in between those
lines would be 30*2=60. This is because there is a two hour difference
between 12 and 2 and there is a 30 degree angle difference between each
hour.
Now, let's look at the minutes. If we draw a line from the centre to
minute 0 (otherwise described as the same position as hour 12) and then
from the centre to minute 36 on the clock edge, we have an angle of
6*36=216.
It would be very simple to calculate the difference between the two
angles however, this is not the angle we are looking for. As the time
changes from 2 and 3 o'clock, the small hour hand shifts between the
hours of 2 and 3, so we need to account for that change in degrees.
Recall that there are 30 degrees between each hour. Then, since 36/60
minutes have passed in that hour, we calculate (36/60)*30=36/2=18
degrees. We then add this to the 60 degree calculated earlier (in bold
above). This is because if we draw a line from the centre to hour 12
and then a line to hour 2 minute 36 then we have an angle of 60+18=78.
Now, we find the difference in 216 and 78. This gives us 216-78=138.
This is our final answer and the angle between the hands of a clock at
2:36. Note that this is the angle from the hour hand to the minute
hand. For the angle between the minute hand to the hour hand, we
subtract 138 from 360 and so, we are left with 360-138=222 degrees.
We can check our answer if by adding the two degrees, 138+222=360, we
obtain a full circle or 360 degrees.