1344. Angle Between Hands of a Clock - Detailed Explanation

Problem Statement

Description:
You are given two integers representing the hour and the minutes on a 12‑hour analog clock. Your task is to compute the smaller angle (in degrees) between the hour hand and the minute hand.

Key Points:

• The clock is a 12‑hour clock.
• The answer should be the minimum angle between the two hands.
• The output is a floating‑point number with an acceptable error margin.

Example 1:

• Input: hour = 12, minutes = 30
• Output: 165.0
• Explanation:
  • The minute hand points at 180° (30 minutes × 6° per minute).
  • The hour hand is at 15° (since 12 is treated as 0, then (0 \times 30 + 30 \times 0.5 = 15)).
  • The absolute difference is (|180 - 15| = 165°).
  • The complementary angle (360 - 165 = 195°) is larger, so the answer is 165°.

Example 2:

• Input: hour = 3, minutes = 0
• Output: 90.0
• Explanation:
  • The minute hand is at 0°.
  • The hour hand is at 90° ((3 \times 30 = 90)).
  • The absolute difference is 90°, and the complementary angle is 270°; hence the result is 90°.

Intuition and Hints

• Understanding Hand Movements:

  • Minute Hand: Moves 360° in 60 minutes. Hence, it moves at 6° per minute.
  • Hour Hand: Moves 360° in 12 hours, but it also moves continuously as minutes pass. Hence, it moves at 30° per hour plus an extra (0.5°) per minute.

• Deriving the Angles:

  • Minute Hand Angle:
    [\text{minuteAngle} = \text{minutes} \times 6]
  • Hour Hand Angle:
    [\text{hourAngle} = (\text{hour} \mod 12) \times 30 + \text{minutes} \times 0.5]
  • Angle Difference:
    [\text{diff} = \left| \text{hourAngle} - \text{minuteAngle} \right|]
  • Minimum Angle:
    Since the hands can form an angle on either side, the smaller angle is: [\min(\text{diff}, 360 - \text{diff})]

• Key Hint:
  Use the derived formulas to compute the two angles and then select the smaller one.

Approach

Approach 1: Direct Formula Calculation

Explanation:

1. Compute the Minute Hand Angle:
   Multiply the number of minutes by 6.

2. Compute the Hour Hand Angle:
   First, take the hour modulo 12 (since 12 o’clock is treated as 0). Multiply by 30 and add the contribution from the minutes ((0.5) degrees per minute).

3. Calculate the Absolute Difference:
   Get the absolute difference between the hour and minute hand angles.

4. Determine the Minimum Angle:
   Since the clock is circular, the smaller angle is the minimum of the difference and (360) minus the difference.

Python Code (Direct Formula Approach):