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Angle Between Hands of a Clock

Difficulty: Medium

Problem Description

Given two numbers, hour and minutes, return the smaller angle (in degrees) formed between the hour and the minute hand.

Key Insights

  • The hour hand moves at a rate of 30 degrees per hour (360 degrees / 12 hours).
  • The minute hand moves at a rate of 6 degrees per minute (360 degrees / 60 minutes).
  • The position of the hour hand also changes slightly as the minutes pass, specifically by 0.5 degrees for each minute (30 degrees / 60 minutes).
  • To find the angle between the two hands, calculate the absolute difference between their positions and ensure to return the smaller angle.

Space and Time Complexity

Time Complexity: O(1)
Space Complexity: O(1)

Solution

To solve the problem, we calculate the position of the hour and minute hands in degrees:

  1. Calculate the position of the hour hand using the formula: hour_position = (hour % 12) * 30 + (minutes * 0.5).
  2. Calculate the position of the minute hand using the formula: minute_position = minutes * 6.
  3. Find the absolute difference between the two positions.
  4. Since the maximum angle in a circle is 360 degrees, the smaller angle can be found by taking the minimum of the calculated angle and 360 - calculated angle.

Code Solutions

def angleClock(hour: int, minutes: int) -> float:
    # Calculate the position of the hour hand
    hour_position = (hour % 12) * 30 + (minutes * 0.5)
    # Calculate the position of the minute hand
    minute_position = minutes * 6
    # Calculate the difference between the two positions
    angle = abs(hour_position - minute_position)
    # Return the smaller angle
    return min(angle, 360 - angle)
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