Formula

Angle between the hands of a clock

Arithmetic • Clocks • Angle Between Hands

Formula

θ = 30 H - \frac{11 M}{2}

Measuring clockwise from the

12

mark, at

H

hours and

M

minutes the minute hand stands at

6 M

degrees and the hour hand at

(30 H + \frac{M}{2})

degrees, so the angle between them is

θ = 30 H - \frac{11 M}{2}

degrees, replaced by

36 0^{\circ} - θ

whenever that value exceeds

18 0^{\circ}

How to use it

1Read $H$ as the hour just passed and $M$ as the minutes, then substitute directly; the $12$ o'clock position counts as $H = 0$ .
2Never leave the hour hand on its numeral: it has already crept $\frac{M}{2}$ degrees past it, and forgetting that is the usual error.
3If the formula returns more than $18 0^{\circ}$ , subtract it from $36 0^{\circ}$ to get the angle normally intended.
4To answer in radians, multiply the result by $\frac{π}{180}$ .

What is the angle between the hour hand and the minute hand at

7

hours

40

minutes?

Step-by-step

1
$θ = 30 \times 7 - \frac{11 \times 40}{2} = ∣ 210 - 220 ∣$
Why: Substitute $H = 7$ and $M = 40$ .
2
$θ = 1 0^{\circ}$
Why: The value is already below $18 0^{\circ}$ , so no reduction is needed.

Answer

1 0^{\circ}

Watch out

The formula measures the angle as a signed difference before the absolute value, so a result above

18 0^{\circ}

is the reflex angle and must be subtracted from

36 0^{\circ}

; and

H

must be taken modulo

12

, since a

24

-hour reading such as

19 : 40

gives

30 H = 570

rather than

210

Also known as

angle between clock hands, clock hand angle, hour and minute hand angle, 30H minus 11M/2, hands of a clock