Angles of a Clock
Adam J. O’Neil
July 19, 2018
Clocks are good tools for teaching angles, in some sense. Unfortunately, clocks
aren’t perfect for this because they violate some mathematical ideals. If mathe-
maticians had built the standard 12-hour analog clock, 12 o’clock would be where
3 o’clock is, and the clock would run counterclockwise (that is somewhat counter-
intuitive, I will admit). At any rate, investigating the movement of the hands on
a clock makes for an interesting display of basic mathematics and problem-solving.
For this example, I will use radians, not degrees, for the initial calculations.
Clock Movement
We are going to consider 0 radians to be located at 12 o’clock, and positive angles
will run clockwise from 0 radians. With this in mind, we can now calculate the
position of the hour hands when 0 minutes and 0 seconds have elapsed in this
current hour – i.e, we can calculate the position of the hour hand at exactly 6:00:00.
Recall that there are 2π radians in one revolution (that is, 2π rad = 360
◦
). This
also means that a single radian is
1
2π
of a revolution, or roughly .1592 revolutions.
To begin, we must notice that there are 12 hours on a standard 12-hour clock,
each taking up 1/12th of the clock (as one would think). These 12 hours also exactly
divide the 2π radians – thus each hour is π/6th of a radian further than the previous
one (
2π
12
=
π
6
).
With such information, we can say that the position of the hour hand at hour
h will be at
hπ
6
. However, this is only entirely true when 0 minutes and 0 seconds
have elapsed into the hour. As the hour progresses, the hour hand will gradually
approach the next hour, changing the angle of this hand. We must contemplate the
movement of the minute and second hands as well.
There are 60 minutes in one hour. Every 60 minutes, we must rotate the hour
hand an additional π/6th of a radian. As such, 1 minute must rotate the hour
hand an additional
π/6
60
of a radian, or π/360 of a radian. Therefore, by letting
m represent the number of minutes that have elapsed in the current hour, we can
write an equation as such (let θ
hr
be the angle of the hour hand).
hπ
6
+
mπ
360
= θ
hr
1
However, our equation is still incomplete, for we have not factored in seconds
yet! There are 60 seconds in one minute, so after 60 seconds have elapsed, the hour
hand should rotate an additional π/360th of a radian – that is, it should rotate such
that an additional minute had just elapsed.. This means that each second should
rotate π/360/60th radian, or π/21600 radian. Letting s represent seconds elapsed
in the hour, we obtain a new equation:
hπ
6
+
mπ
360
+
sπ
21600
= θ
hr
Which accurately describes the position of the hour hand to the nearest second
interval. We should probably factor the equation, however:
1
6
[hπ +
mπ
60
+
sπ
3600
] = θ
hr
Factoring makes the equation only marginally easier to work with. We could
instead work with an equation that works with degrees instead, even though degrees
are inferior to radians, they are easier to picture in one’s head. Most people can
picture an angle of 75 degrees easier than they can imagine an angle of 1.31 radians.
We can easily convert to degrees by multiplying each term by 180/π.
30h
◦
+ (
m
2
)
◦
+ (
s
120
)
◦
= θ
hr.deg
When one considers it, the degree version of this equation is a little better-
looking, albeit less useful.
Okay, now let’s calculate the position of the minute hand at a given time. The
hour information is irrelevant; the only other variable that affects the minute hand
is where the second hand currently is. There are 60 minutes in an hour, and it
takes 60 minutes for the minute hand to make one revolution around the clock.
Since there are 2π radians in 60 minutes, each minute must be π/30th of a radian.
For the second hand, it must make the minute hand rotate π/30th of a radian at
60 seconds, so each single second will make the minute hand rotate π/1800th of a
radian. Letting m represent minutes elapsed, and s representing seconds elapsed,
we obtain the following equations:
mπ
30
+
sπ
1800
= θ
min
1
30
[mπ +
sπ
60
] = θ
min
Equivalently, in degrees:
6m
◦
+
s
10
◦
= θ
min.deg
2
Finally, let us consider the position of the easiest hand: the second hand, which
is not dependent upon other hands in any way. It makes one revolution every
60 seconds, which means that each second rotates the second hand π/30 radians.
Again, allowing s to represent seconds elapsed, we obtain:
sπ
30
= θ
sec
6s
◦
= θ
sec.deg
That is all there is to it. We now have precise angle calculations for all three
hands; we could, for example, put these equations to work by calculating angles
between hands, which is now a trivial exercise that would just involve adding these
equations together in some manner.
Here are the factored equations:
1
6
[hπ +
mπ
60
+
sπ
3600
] = θ
hr
(1)
1
30
[mπ +
sπ
60
] = θ
min
(2)
sπ
30
= θ
sec
(3)
Alternatively, if you prefer (I like this way better), these are the same equations
with a single denominator, which is easier if you are working without a calculator:
3600hπ + 60mπ + sπ
21600
= θ
hr
(4)
60mπ + sπ
1800
= θ
min
(5)
sπ
30
= θ
sec
(6)
Example
- Find the angle of the hour, minute, and second hands at 2:31:14. In variables, h
= 2, m = 31, s = 14. For this example, I will rewrite the above equations such that
we only have a single denominator multiplied by a constant.
3600hπ + 60mπ + π
21600
= θ
hr
7200π + 1860π + 14π
21600
= θ
hr
9074π
21600
= θ
hr
≈ 1.3198 rad
3
The hour hand is rotated about 1.3198 rad, or 75.6
◦
. For the minute and second
hands, I will similarly rewrite as a single denominator.
60mπ + sπ
1800
= θ
min
1860π + 14π
1800
= θ
min
1874π
1800
= θ
min
≈ 3.2707 rad
Finally, the easiest calculation, which is the rotation of the second hand:
14π
30
= θ
sec
≈ 1.466 rad
In degrees, the minute hand is rotated roughly 187.4
◦
and the second hand about
84.0
◦
.
Sigma
This interesting exercise illustrates a situation where the position of one object is
dependent upon the position of another. In this case, of course, we are dealing
with angles, not necessarily the position of an object in three-dimensional space,
but regardless of that it is the same principle at work.
- AJO
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