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Clock Angle

Calculate the angle between the hour and minute hands on an analog clock for any time.

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Why: Understanding clock angle helps you make better, data-driven decisions.

How: Enter Hour (0-23), Minute (0-59) to calculate results.

Run the calculator when you are ready.

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Clock Angle — Hour & Minute Hand

Find the angle between clock hands at any time. Hour hand: 30° per hour + 0.5° per minute.

⏰ Sample Times — Click to Load

Input

3:30
75.00°
clock_angle.shCALCULATED
$ clock_at 3:30
Time
3:30
Angle
75.00°
Hour hand
105.00°
Minute hand
180.00°
Share:
Clock Angle
3:30
75.00°
numbervibe.com/calculators/mathematics/angle/clock-angle-calculator

Angle Distribution

Hand Positions

Example Times

📐 Calculation Breakdown

INPUT
Given time
3:30
CALCULATION
Hour hand position
105.00°
\text{theta} _h = 30H + 0.5M
CALCULATION
Minute hand position
180.00°
\text{theta} _m = 6M
CALCULATION
Raw difference
75.00°
|\text{theta} _h - \text{theta} _m|
RESULT
Smaller angle
75.00°
\text{min}( ext{angle}, 360^{circ} - ext{angle})

For educational and informational purposes only. Verify with a qualified professional.

📋 Key Takeaways

  • • Hour hand: 30° per hour + 0.5° per minute (moves continuously)
  • • Minute hand: 6° per minute
  • • Formula: angle = |30H - 5.5M|, then take min(angle, 360° - angle)
  • • At 12:00 exactly, both hands align (0°)
  • • At 3:15, the angle is 7.5° — not 90° (common misconception)

💡 Did You Know?

🕐The hands overlap exactly 11 times in a 12-hour period — not at each hour mark!Source: Wolfram MathWorld
📐A right angle (90°) occurs 22 times in 12 hoursSource: Math is Fun
⏱️The hour hand moves 0.5° per minute because 30° ÷ 60 = 0.5°Source: Khan Academy
🔄The minute hand moves 6° per minute (360° ÷ 60)Source: GeeksforGeeks

📖 How It Works

Hour hand: θ_h = 30H + 0.5M. Minute hand: θ_m = 6M. The angle between them is |θ_h - θ_m|. We take the smaller angle (min of result and 360° - result).

🎯 Expert Tips

💡 Use 12-hour format

Hours 0–23 auto-convert to 12-hour (0–11).

💡 3:15 ≠ 90°

At 3:15 the hour hand has moved 7.5° past 3, so the angle is 7.5°.

⚖️ Comparison Table

TimeAngle
12:000°
3:0090°
6:00180°
9:0090°
3:157.5°

❓ Frequently Asked Questions

At what times do the hands overlap?

Approximately 11 times in 12 hours: 12:00, ~1:05:27, ~2:10:55, etc.

Why 0.5° per minute for the hour hand?

The hour hand moves 30° per hour. 30° ÷ 60 minutes = 0.5° per minute.

Is 3:15 a right angle?

No. At 3:15 the hour hand is at 97.5°, minute at 90°. The angle is 7.5°.

📊 Infographic Stats

30°
Per hour
6°
Per minute (min hand)
0.5°
Per minute (hour hand)
11
Overlaps in 12h

⚠️ Disclaimer: This calculator provides educational estimates. Seconds are not considered. For precision engineering use authoritative sources.

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