1344. Angle Between Hands of a Clock

1344. Angle Between Hands of a Clock

Difficulty: Medium

Topics: Staff, Math, Biweekly Contest 19

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

Answers within 10⁻⁵ of the actual value will be accepted as correct.

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

Hint:

1. The tricky part is determining how the minute hand affects the position of the hour hand.
2. Calculate the angles separately then find the difference.

Solution:

We implement a direct mathematical solution to compute the smaller angle between the clock hands. By calculating each hand’s angular position independently and then taking the minimum of the absolute difference and its complement to 360°, we efficiently obtain the result in constant time.

Approach

• Minute hand angle – Each minute moves the minute hand by 6° (since 360° / 60 = 6°).
• Hour hand angle – Each hour moves the hour hand by 30° (since 360° / 12 = 30°), plus an additional 0.5° per minute (since 30° / 60 = 0.5°).
• Normalize hour – Convert 12 to 0 because the 12‑hour dial wraps around.
• Compute difference – Take the absolute angular difference.
• Return smaller angle – Since two lines form two angles (θ and 360°−θ), we return the minimum.

Let's implement this solution in PHP: 1. Angle Between Hands of a Clock

<?php
/**
 * @param Integer $hour
 * @param Integer $minutes
 * @return Float
 */
function angleClock(int $hour, int $minutes): float
{
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

// Test cases
echo angleClock(12, 30);        // Output: 165
echo angleClock(3, 30);         // Output: 75
echo angleClock(3, 15);         // Output: 7.5
?>

Explanation:

• The minute hand angle is simply minutes * 6.
• The hour hand angle is (hour % 12) * 30 + minutes * 0.5, accounting for the hour hand's continuous movement.
• The absolute difference |hourAngle - minuteAngle| gives one of the two angles.
• The smaller angle is always min(diff, 360 - diff) because the total circle is 360°.
• This works for all valid inputs (1–12 hour, 0–59 minutes) and meets the required precision (10⁻⁵).

Complexity Analysis

• Time Complexity: O(1) – only a few arithmetic operations, independent of input size.
• Space Complexity: O(1) – uses only a constant number of variables.

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