Time for angles

04 March, 2019Readtime: 3 mins

I was once asked in an interview to calculate the angle between the minute and hour hands on a clock for the time 16:34. This was tricky to do in my head but possible given 2 simple facts.

The minute hand moves 6 degrees for every minute, since we have 360 degrees for one revolution and 60 minutes in an hour, hence 360/60 = 6 degrees/minute.
The second is that the hour hand moves 30 degrees for every hour, again 360 degrees and now 12 hours in one revolution, hence 360/12 = 30 degrees/hour.

However, the hour hand also moves 0.5 degrees for every minute, since the hour hand covers 30 degrees in 60 minutes. Well knowing this we can then simply do 34 X 6 degrees = 204 degrees for the minute hand and (4 X 30 degrees) + (34 X 0.5 degrees) = 137 degrees for the hour hand. Thus, we have (taking the small angle) a difference of 67 degrees, or 1.17 radians if you prefer.

The obvious follow up question was then - What is the time when the hands next cross? Well to answer this I have plotted the angle as a function of time (in minutes).

We can see from printing out the times at which the angle is a minimum gives us 17:27 (to the nearest minute). The full list of cross over times are:

Time(HH:MM)	Angle (Deg)
00:00	0.0
01:05	2.5
02:11	0.5
03:16	2.0
04:22	1.0
05:27	1.5
06:33	1.5
07:38	1.0
08:44	2.0
09:49	0.5
10:55	2.5
12:00	0.0
13:05	2.5
14:11	0.5
15:16	2.0
16:22	1.0
17:27	1.5
18:33	1.5
19:38	1.0
20:44	2.0
21:49	0.5
22:55	2.5

Alas, I didn’t get the job in the end, but for anybody who has this question, or is simply curious next time they are staring at the clock, the vanilla Python script (version 3 compatible) below should help.

import math

#constants
twoPi = 2.0*math.pi
maxDegree = 360.0
secondsInHour = 60
minutesInHour = secondsInHour
hoursInDay = 24
maxHour = math.ceil(hoursInDay/2)

class Clock():
    def __init__(self, hour, minute, radians):
        self.hour = hour
        self.minute = minute
        self.radians = radians
        self.factor = twoPi
        if not radians:
            self.factor = maxDegree
        
        self.validateTime()
    
    def validateTime(self):
        if self.hour > hoursInDay - 1 or self.hour < 0:
            raise ValueError('Time must be valid - check hours')
        
        if self.minute > minutesInHour - 1 or self.minute < 0:
            raise ValueError('Time must be valid - check minutes')
                        
    def __str__(self):
        return "Standard time is: {}, Time in minutes : {}, , Angle: {}".format( 
            self.getTimeStandardAsString(), 
            self.getTimeInMinutes(), 
            self.computeAngle())            
    
    def getTimeStandardAsString(self):
        timeInMinutes = self.getTimeInMinutes()
        hours = str(math.floor(timeInMinutes/minutesInHour))
        mins = str(timeInMinutes % minutesInHour)
    
        if len(hours) == 1:
            hours = "0" + hours
            
        if len(mins) == 1:
            mins = "0" + mins
            
        return hours + ":" + mins
    
    def getTimeInMinutes(self):
        return self.hour*minutesInHour + self.minute
    
    # every minute moves the minute hand by 6 degrees
    def computeMinuteAngle(self):
        return self.minute*self.factor/minutesInHour
    
    # every hour moves the hour hand by 30 degrees and each minute moves the hour hand by 0.5 degrees
    def computeHourAngle(self):
        # convert to non 24 hour (between 0 and 12 only)
        simpleTimeHour = self.hour
        if simpleTimeHour > maxHour:
            simpleTimeHour = simpleTimeHour - maxHour
    
        return (simpleTimeHour*self.factor + self.computeMinuteAngle())/maxHour
        
    def computeSmallAngle(self):
        minuteAngle = self.computeMinuteAngle()
        hourAngle = self.computeHourAngle()
    
        angle = abs(hourAngle - minuteAngle)
        
        return min(angle, abs(self.factor - angle))
    
    def computeLargeAngle(self):
        return self.factor - self.computeSmallAngle()
    
    def computeAngle(self):
        return self.computeSmallAngle()

If we then plug in the desired time, 16:34, behold we get 67 deg.

clock = Clock(16,34, False)
print(clock)

Written by Thomas Stainer who likes to develop software for applications mainly in maths and physics, but also to solve everyday problems. Check out my GitHub page here.