Angle Between Hands of a Clock

Description:​

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

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

Video Explanation​

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Approaches:​

1. Mathematical Approach (Relative Speeds)​

To find the angle between the hands, we calculate the exact position (in degrees) of both the minute and hour hands relative to the 12:00 position, and then find the absolute difference between them.

1. Minute Hand: The minute hand moves $360^\circ$ in $60$ minutes. Therefore, it moves at a speed of $6^\circ$ per minute.
2. Hour Hand: The hour hand moves $360^\circ$ in $12$ hours ($720$ minutes). Therefore, it moves at a speed of $0.5^\circ$ per minute. It also shifts based on the current hour.
3. Find the Difference: Subtract the two angles and take the absolute value.
4. Find the Smaller Angle: A clock is $360^\circ$. If the difference is greater than $180^\circ$, the smaller angle will be on the other side. Return the minimum of the difference and $360$ minus the difference.

• Time Complexity: $O(1)$ because the calculation relies on a few basic arithmetic operations, regardless of the input size.
• Space Complexity: $O(1)$ because we only use a few variables to store the angles, requiring constant extra space.

Solutions​

class Solution {
public:
    double angleClock(int hour, int minutes) {
        double minute_angle = minutes * 6.0;
        double hour_angle = (hour % 12) * 30.0 + minutes * 0.5;
        
        double diff = std::abs(hour_angle - minute_angle);
        
        return std::min(diff, 360.0 - diff);
    }
};

class Solution {
    public double angleClock(int hour, int minutes) {
        double minuteAngle = minutes * 6.0;
        double hourAngle = (hour % 12) * 30.0 + minutes * 0.5;
        
        double diff = Math.abs(hourAngle - minuteAngle);
        
        return Math.min(diff, 360.0 - diff);
    }
}

class Solution:
    def angleClock(self, hour: int, minutes: int) -> float:
        minute_angle = minutes * 6
        hour_angle = (hour % 12) * 30 + minutes * 0.5
        
        diff = abs(hour_angle - minute_angle)
        
        return min(diff, 360 - diff)

/**
 * @param {number} hour
 * @param {number} minutes
 * @return {number}
 */
const angleClock = function(hour, minutes) {
    const minuteAngle = minutes * 6;
    const hourAngle = (hour % 12) * 30 + minutes * 0.5;
    
    const diff = Math.abs(hourAngle - minuteAngle);
    
    return Math.min(diff, 360 - diff);
};