Calculating the Angle Between Hour and Minute Hands Using Python

In this blog, you will learn how to calculate the angle between the hour and minute hands of a clock using Python. The solution is based on a simple mathematical approach explained below.

At 12:00, both the hour and minute hands align at the 12 o’clock position. To calculate the angle at any given time:

1. Calculate the “hour hand angle” and the “minute hand angle” relative to the 12 o’clock position.
2. Take their absolute difference to determine the angle between the two hands.
3. If you need the minimum angle, compare the absolute difference with 360°- absolute difference and take the smaller value.

Here are the key formulas:

Hour Hand Angle: The hour hand moves 30° per hour and 0.5° per minute. Therefore,
Hour Hand Angle = (H*30) + (M*0.5), where H is the hour, and M is the minutes past the hour.

Minute Hand Angle: The minute hand moves 6° per minute. Therefore, Minute Hand Angle = M*6

The angle between the hands is:
Absolute Angle = ∣Hour Hand Angle – Minute Hand Angle∣

To run the following Python code, provide the time as a command-line argument in the following format:

python calc_angle.py 8:15

Here, calc_angle.py is the name of the program containing the code. If you use a different filename, replace calc_angle.py with your program’s name. 8:15 represents the time for which you want to calculate the angle between the hour and minute hands.

import sys

def validate_time(user_time):
    """
    check if user provided correct time
    """
    hrs, mins = int(user_time.split(':')[0]), int(user_time.split(':')[1])
    if 0 <= hrs <= 12 and 0 <= mins <= 59:
        return True, hrs, mins
    else:
        return False, hrs, mins



def calculate_angle(hour, minute):
    """
    Find angle between hour and minute hands
    """
    # Calculate the position of the hour hand in degrees w.r.t 12
    hour_angle = (hour * 30) + (minute * 0.5)

    # Calculate the position of the minute hand in degrees w.r.t 12
    minute_angle = minute * 6

    # Calculate the absolute difference between the two angles
    angle = abs(hour_angle - minute_angle)

    # The smaller angle between the two hands
    return min(angle, 360 - angle)


def main():
    """
    This program calculates the angle between minute and hour hands for a given time
    """
    # provide time as command-line argument, e.g. 10:45
    args = sys.argv

    # check if user provided time or not
    if len(args) < 2:
        print("No time provided. please enter time (e.g. 9:45) as command-line argument")
    else:
        # print("time provided by user: ", args[1])

        # validate time provided by user
        correct_time, hh, mm = validate_time(args[1])
        if correct_time:
            angle = calculate_angle(hh, mm)
            print(f"the angle between the hour and minute hands at {args[1]} is {angle} degrees.")
        else:
            print(f"{args[1]} is invalid time")

if __name__ == "__main__":
    main()

Please let me know in the comments if you find any errors in the code.