Calculate the angle between the hour and minute hands of an analog clock. Shows reflex angle, hand positions, and a clock face visual with a full-day angles table.
The clock angle problem is a classic math puzzle: given a time, what is the angle between the hour and minute hands of an analog clock? The minute hand moves at 6° per minute (360°/60), while the hour hand moves at 0.5° per minute (360°/720). At any time, the absolute angle between the hands is |30H − 5.5M| degrees, where H is the hour (1–12) and M is the minutes. If this value exceeds 180°, the smaller angle is 360° minus the result. This calculator goes further: it shows both the acute/obtuse and reflex angles, the exact positions of each hand in degrees from 12 o'clock, the angular velocity of each hand, and the next times the hands overlap or are exactly opposite. A clock face SVG diagram updates in real time, and a comprehensive table shows angles at every 15-minute mark throughout a 12-hour period. The clock angle problem appears frequently in competitive math, job interviews, and standardized tests. It also has practical applications in sundial design, compass navigation, and astronomical calculations. Presets cover famous examples like 3:00 (90°), 6:00 (180°), and tricky times like 9:49 where the angle is surprisingly small.
The clock angle formula |30H − 5.5M| is simple, but tricky cases arise when the result exceeds 180° (requiring the reflex correction) or when minutes cause the hour hand to shift. This calculator handles all edge cases, shows both the smaller and reflex angles, visualizes the exact hand positions on an SVG clock face, and computes when the hands next overlap or become opposite. It is perfect for math competition prep, job interview puzzles, and teaching angular velocity concepts.
Angle = |30H − 5.5M| If Angle > 180, use 360 − Angle Minute hand: 6° per minute Hour hand: 0.5° per minute
Result: 7.50°
At 3:15: |30×3 − 5.5×15| = |90 − 82.5| = 7.5°. At 9:00: |30×9 − 5.5×0| = 270° → 360 − 270 = 90°.
The minute hand completes 360° in 60 minutes, so it moves at 6° per minute. The hour hand completes 360° in 12 hours (720 minutes), so it moves at 0.5° per minute. At time H:M, the minute hand is at 6M degrees from 12, and the hour hand is at 30H + 0.5M degrees. The angle between them is |30H − 5.5M|. If this exceeds 180°, take 360° minus the result to get the smaller angle. The key insight students often miss is the 0.5M term — the hour hand does not jump from hour to hour but moves continuously.
At 3:00, the angle is exactly 90° — one of the few times the answer is a round number. At 6:00, the hands are opposite (180°). At 12:00, they overlap (0°). Trickier problems include 3:15 (only 7.5° because the hour hand has moved past 3) and 9:49 (the hands are nearly overlapping at about 5.5°). Competition problems often ask: at what times are the hands exactly 90° apart, or when do they overlap? The hands overlap 11 times in 12 hours (approximately every 65.45 minutes), and they are opposite 11 times as well.
The relative angular velocity of the minute hand with respect to the hour hand is 5.5° per minute (6° − 0.5°). This means the minute hand "laps" the hour hand every 360°/5.5° ≈ 65.45 minutes. This relative motion concept connects clock problems to physics topics like orbital mechanics and gear ratios. In engineering, the same math applies to rotating machinery where two shafts turn at different speeds and periodically align.
Use the formula |30H − 5.5M| where H is the hour (1–12) and M is the minutes. If the result exceeds 180°, subtract from 360°.
At 3:00, the angle is |30×3 − 5.5×0| = 90°.
The reflex angle is the larger of the two angles formed by the hands (greater than 180°). It equals 360° minus the smaller angle.
The hands overlap approximately every 65.45 minutes, or exactly 11 times in a 12-hour period (at 12:00, ~1:05, ~2:11, etc.).
The hands are opposite at 6:00 and approximately every 65.45 minutes thereafter (10 more times in 12 hours, avoiding exact half hours).
This calculator focuses on hour and minute hands. The second hand moves at 6° per second but is typically ignored in clock angle problems.