About

The angle between clock hands is a deceptively tricky geometric problem. The hour hand moves at 0.5°/min, not in discrete jumps. A common error is treating the hour hand as stationary between hours. At 3:00 the angle is exactly 90°, but at 3:15 the hour hand has already swept an additional 7.5°, making the true angle 7.5° rather than the 0° many assume. This calculator resolves the continuous positions of all three hands using degree-per-second rates and always returns the smaller of the two possible arcs. Precision matters in competitive math, interview questions, and mechanical watchmaking tolerances where fractions of a degree affect gear ratios.

The tool assumes an ideal clock with perfectly uniform hand motion. Real mechanical clocks exhibit backlash in gear trains and quartz movements advance the minute hand in discrete steps. For quartz clocks, set seconds to 0 and treat the minute hand as jumping in whole-minute increments. The second hand contribution to the hour hand position (1⁄120° per second) is included for completeness but is negligible for most practical purposes.

Formulas

All angles are measured clockwise from the 12 o'clock position. The hour hand advances continuously, not in discrete jumps.

θ_h = (h mod 12) × 30 + m × 0.5 + s × 1120

θ_m = m × 6 + s × 0.1

θ_s = s × 6

δ = |θ₁ − θ₂|

angle = min(δ, 360 − δ)

Where h = hours (0 - 23, converted to 12-hour), m = minutes (0 - 59), s = seconds (0 - 59). The coefficient 30 arises from 360°12 hours, the coefficient 6 from 360°60 minutes, and 0.5 from 30°60 minutes. The final min operation guarantees the result is always the smaller arc, bounded by [0°, 180°].

Reference Data

Time	Hour Hand (θ_h)	Minute Hand (θ_m)	Angle Between	Relation
12:00	0°	0°	0°	Overlapping
1:00	30°	0°	30°	Acute
2:00	60°	0°	60°	Acute
3:00	90°	0°	90°	Right angle
3:15	97.5°	90°	7.5°	Acute
4:00	120°	0°	120°	Obtuse
5:00	150°	0°	150°	Obtuse
6:00	180°	0°	180°	Straight line
7:00	210°	0°	150°	Obtuse (reflex taken)
8:00	240°	0°	120°	Obtuse (reflex taken)
9:00	270°	0°	90°	Right angle
10:00	300°	0°	60°	Acute (reflex taken)
11:00	330°	0°	30°	Acute (reflex taken)
12:30	15°	180°	165°	Obtuse
1:05	32.5°	30°	2.5°	Near overlap
5:27	163.5°	162°	1.5°	Near overlap
6:30	195°	180°	15°	Acute
9:15	277.5°	90°	172.5°	Obtuse
10:10	305°	60°	115°	Obtuse
2:45	82.5°	270°	172.5°	Obtuse
11:55	357.5°	330°	27.5°	Acute

Frequently Asked Questions

The hour hand on an analog clock completes a full 360° rotation in 12 hours, giving it a continuous rate of 0.5° per minute. At 3:30, for example, the hour hand is not sitting on the 3 but has advanced to 105° (90° + 15°). Ignoring this continuous motion produces errors up to 29.5° - nearly an entire hour mark of deviation.

The hands overlap 11 times in 12 hours, not 12. The minute hand laps the hour hand once per ~65.4545 minutes (the hands share the 12:00 overlap at both start and end of the cycle, so it counts as one). The overlap times can be computed by solving θ_h = θ_m, yielding intervals of 720/11 ≈ 65.4545 minutes.

The second hand contributes 0.1° per second to the minute hand position and 1/120° per second to the hour hand. At maximum (s = 59), this shifts the minute hand by 5.9° and the hour hand by ~0.49°. For exam-style problems that specify only hours and minutes, set seconds to 0. For precise mechanical watchmaking, include seconds.

In 12 hours, the hands form a 90° angle exactly 22 times (11 times each for 90° and 270° raw difference). The first occurrence after 12:00 is at approximately 12:16:21.8. These times are not evenly spaced - consecutive right-angle events alternate between intervals of roughly 32.7 and 32.7 minutes.

The raw angular difference can reach up to 360°, but by convention the "angle between" two clock hands refers to the smaller of the two arcs formed. This calculator always returns min(δ, 360° − δ), capping the result at 180°. A result of exactly 180° means the hands are diametrically opposite (e.g., at 6:00:00).

The calculator accepts hours from 0 to 23. Internally, h mod 12 converts to the 12-hour cycle. 0:00 and 12:00 both map to the 12 o'clock position. 13:00 maps identically to 1:00. The displayed clock face always shows the 12-hour analog representation regardless of input format.