Angle Between Hands of a Clock

Problem Statement

The challenge involves calculating the smaller angle in degrees between the hour and minute hands on a traditional clock based on given "hour" and "minutes" input values. The objective is to determine the lesser of the two potential angles that could be formed from the 12 o'clock position. This calculation must consider the continuous movement of the hour hand as minutes pass, which also influences the hour hand's position relative to the minute hand. Accuracy is essential, as the result must be within a very small tolerance of the exact angle.

Examples

Example 1

Input:

hour = 12, minutes = 30

Output:

165

Example 2

Input:

hour = 3, minutes = 30

Output:

75

Example 3

Input:

hour = 3, minutes = 15

Output:

7.5

Constraints

• 1 <= hour <= 12
• 0 <= minutes <= 59

Approach and Intuition

Calculating the smaller angle between the hour and minute hands on a clock involves understanding the movement dynamics of these hands:

1. Minute Hand Movement: The minute hand moves by 360 degrees over 60 minutes, which is 6 degrees per minute.
2. Hour Hand Movement: The hour hand, covering 360 degrees over 12 hours, moves 30 degrees per hour. In addition, it progresses further as the minutes pass, at a rate of 0.5 degrees per minute (since 30 degrees/hour over 60 minutes/hour equals 0.5 degrees/minute).

Steps to Compute the Angle

1. Calculate Minute Hand Angle: This is simply 6 degrees multiplied by the number of minutes past the hour.
2. Calculate Hour Hand Angle: This is 30 degrees multiplied by the hour, plus 0.5 degrees multiplied by the minutes.
3. Compute the Difference: The absolute difference between the angles gives the larger of the two possible angles formed.
4. Determine the Smaller Angle: Since a full circle is 360 degrees, subtract the larger angle from 360 if the initial difference is greater than 180 degrees, as the hands could form a smaller angle going the other way round the dial.

These steps will yield the smaller of the two possible angles between the hour and minute hands of a clock, adjusted correctly for the movement of both hands, while considering only valid input ranges based on clock mechanics.

Solutions

class Solution {
  public:
  double findClockAngle(int h, int m) {
    int anglePerMinute = 6;
    int anglePerHour = 30;
    
    double minutesPosition = anglePerMinute * m;
    double hoursPosition = (h % 12 + m / 60.0) * anglePerHour;
    
    double angleDifference = abs(hoursPosition - minutesPosition);
    return min(angleDifference, 360 - angleDifference);
  }
};