Clock Angle Calculator: Angle Between the Hour and Minute Hands

How it works

Hour hand = 30° × H + 0.5° × M (+ S ÷ 120) · minute hand = 6° × M (+ 0.1° × S) · angle = |30H − 5.5M|, and if it is over 180° use 360° − angle · times for an angle θ: M = (30H ± θ) ÷ 5.5, or t = (θ + 360k) ÷ 5.5 and (360 − θ + 360k) ÷ 5.5 minutes after 12:00

A clock face is 360°, so each hour mark is 30° apart. The minute hand turns 360° in 60 minutes, 6° per minute. The hour hand turns 30° per hour, which is 0.5° per minute, so at 3:40 it is not on the 3 but two thirds of the way to the 4. Measure both hands clockwise from 12, subtract, and take the smaller of the difference and 360° minus it. The two steps fold into the well-known shortcut |30H − 5.5M|. Seconds can be added too: they move the minute hand 0.1° and the hour hand 1/120° per second. The second mode works backwards. Because the minute hand gains 5.5° per minute on the hour hand, the gap repeats a given angle twice an hour on average: 22 times in 12 hours for most angles, and 11 times for 0° (hands together) and 180° (hands opposite). The table lists every time, rounded to the second, and the exact time with the minute as a fraction of 11, as puzzle books give it. This is a geometry tool, not a time-card tool: the clock-out-time and clock-time-to-decimal calculators handle working hours.

Worked example

At 3:40 the hour hand is at 30° × 3 + 0.5° × 40 = 110° from 12 and the minute hand is at 6° × 40 = 240°. The difference is |110 − 240| = 130°, which is under 180°, so the hands are 130° apart (an obtuse angle, 2.2689 radians) and 230° apart the other way round. The shortcut gives the same: |30 × 3 − 5.5 × 40| = |90 − 220| = 130°. The hands next overlap at 4:21:49 and are next opposite at 3:49:05.

Sources: Omni Calculator: Clock angle (10:14 and 8:23 worked examples) · Cache Sleuth: Clock angle tool (|30H − 5.5M| and angle-to-time mode)

Frequently asked questions