Concepts · Clocks

The clock angle

The hour hand moves too. One line counts it for you.

Riya has a clock question in a mock test:

What is the angle between the hands of a clock at 3:40?

She pictures it: the minute hand on the 8, the hour hand on the 3. That is five hour marks apart, at 30° each: 150°. 150° is one of the options, and she ticks it.

The kitchen clock

At home the kitchen clock has stopped at three o’clock. Riya puts in a new battery and sets it to the right time, 3:40, by turning the little knob at the back. The minute hand sweeps round from the 12 to the 8.

And the short hand moves with it. It has crept off the 3, most of the way towards the 4.

Of course it has, she thinks. At half past three, the hour hand is halfway between 3 and 4. At 3:40 it is two-thirds of the way. The hour hand takes a whole hour, 60 minutes, to move one hour mark, 30°, so it moves half a degree a minute. In 40 minutes it has moved 20°. Her 150° in the mock was 20° too many.

The rule

At H hours and M minutes, the angle between the hands is 30 × H − 5.5 × M, without its sign. If that comes to more than 180°, take it from 360° for the smaller angle.

Where it comes from: the hour marks are 30° apart (360 ÷ 12). The minute hand moves 6° a minute (360 ÷ 60), so it is 6 × M from the 12. The hour hand is 30 × H from the 12, plus half a degree for every minute. The gap is 30 × H + 0.5 × M − 6 × M, which is 30 × H − 5.5 × M. Read H on the 12-hour dial: at 15:40, H is 3.

Riya’s question, again

1. 30 × 3 = 90, and 5.5 × 40 = 220.
2. 90 − 220 = −130. Drop the sign: 130. That is less than 180, so the angle is 130°.

More than 180°

At 9:10: 30 × 9 = 270, and 5.5 × 10 = 55. 270 − 55 = 215, which is more than 180. The hands make two angles that add up to 360°. “The angle between the hands” means the smaller one (unless the question asks for the reflex angle), so here it is 360 − 215 = 145°.

Why it helps in the paper

Leaving the hour hand on its number gives a whole number of hour marks, a multiple of 30° like Riya’s 150°, and that answer can sit among the options. The line 30 × H − 5.5 × M counts the creep every time, and takes two multiplications.

Try three

Next mock

A week later, at 4:20, Kabir glances at the classroom clock. The minute hand is on the 4, and he says the hands are together. Riya writes the line instead: 30 × 4 = 120, 5.5 × 20 = 110. The hour hand has crept on past the 4. “Not quite,” she says. “10° apart.”