AFCAT Numerical Ability · Clock

The Angle Between the Hands at a Given Time

Place each hand in degrees from 12 and subtract: the shortcut is the size of 30 times the hour minus 5.5 times the minutes, and the answer is the smaller of the two angles.

All Clock notes

Why this matters

This is the chapter's largest page and mostly EASY: a time is given and the angle is asked. Marks go in two places, forgetting that the hour hand has moved past its number, and giving the angle the long way round when the formula comes out above 180°.

Concept 1 of 4: The angle on the hour

On the hour the minute hand points at 12 and the hour hand sits exactly on its number. Neighbouring numbers are

30°

apart, so count the gaps the short way round and multiply by 30.

Definition

At $h$ o'clock the angle is $30h$ , or $360 - 30h$ when $30h$ is more than 180.
The short way from 9 to 12 is 3 gaps, so 9:00 is $90°$ , the same as 3:00.
6:00 is the only hour when the hands form a straight angle, and 12:00 the only one when they coincide.

On the hour

\theta = 30h \;\text{ or }\; 360 - 30h

Worked example

Find the angle between the hands at 5:00 and at 10:00.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Clock · Angle Between the Hands

What is the angle between the hands of the clock at 4:00?

Count the short way

At 11:00 the hour hand is 11 gaps from 12 one way but only 1 the other. The angle between the hands is

30°

, not

330°

Drill 8 more on the angle on the hour

Concept 2 of 4: The angle at any time

Place each hand by how far it has turned from 12: the hour hand

30h + 0.5m

, the minute hand

6m

. The angle is the difference, and the two combine into one line,

|30h - 5.5m|

Definition

Hour hand at $h{:}m$ : $30h + 0.5m$ degrees from 12. Minute hand: $6m$ degrees.
Angle $= |30h - 5.5m|$ . The 5.5 is the relative speed from the first page.
At half past, the hour hand is exactly halfway between two numbers, $15°$ past the first.
Each minute changes the angle by $5.5°$ , so a minute misread moves the answer by $5.5°$ .

Angle between the hands

\theta = |30h - 5.5m|

Worked example

Find the angle between the hands at 3:40.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Clock · Angle Between the Hands

At 2:38, what is the exact angle between the hands?

The hour hand has moved

At 5:40 the hour hand is

20°

past the 5, at

170°

. Taking it at

150°

gives

90°

instead of the true

70°

Half past is not on the number

At 7:30 the hour hand is halfway between 7 and 8, at

225°

. The angle is

|210 - 165| = 45°

, not

30°

Drill 20 more on the angle at any time

Concept 3 of 4: Above 180°: the smaller angle and the reflex angle

Two hands make two angles that add up to

360°

. A question means the smaller one unless it says reflex. When

|30h - 5.5m|

comes out above 180, subtract it from 360.

Definition

Smaller angle $= \theta$ if $\theta \le 180$ , otherwise $360 - \theta$ .
Reflex angle $= 360 -$ the smaller angle, always above $180°$ .
At 12 and some minutes, take $h = 0$ or $h = 12$ : one gives each of the two angles, and the smaller is the answer either way.

Smaller and reflex angle

\theta_{\text{small}} = \min(\theta,\ 360 - \theta), \qquad \theta_{\text{reflex}} = 360 - \theta_{\text{small}}

Worked example

Find the angle between the hands at 11:05, and the reflex angle.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Clock · Angle Between the Hands

What is the reflex angle between the hands of a clock at 10:40?

Above 180 is the long way round

At 12:44 the formula gives

242°

. The hands are

118°

apart the short way;

242°

is the reflex angle, the answer only when the question asks for it.

Reflex is 360 minus the smaller angle

If the smaller angle is

70°

, the reflex angle is

290°

. Adding 180 gives

250°

, which is neither angle.

Drill 4 more on above 180°: the smaller angle and the reflex angle

Concept 4 of 4: Naming, comparing and changing angles

These questions need two facts, not one: an angle and its type, or the angles at two times. Work each angle with the formula, then name it, compare the two, or subtract.

Definition

Acute: under $90°$ . Right: exactly $90°$ . Obtuse: between $90°$ and $180°$ . Straight: $180°$ .
Which time has the smaller angle: find both angles. Times that look alike can give very different angles.
The change in the angle between two times is the difference of the two angles.

Change in the angle

\Delta\theta = |\theta_2 - \theta_1|

Worked example

Is the angle between the hands at 4:40 acute or obtuse? By how many degrees does the angle change from 8:05 to 8:25?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionHard

Example 4 · Clock · Angle Between the Hands

From

5:10

5:40

, by how many degrees does the angle between hands change?

Work the angles, do not judge the times

6:15 and 6:45 look alike, but the angles are

97.5°

and

67.5°

: one obtuse, one acute.

Near 90 still needs the exact value

An angle of

87.5°

is acute even though it is close to a right angle. Name the type from the worked value, never from a rough picture.

Drill 7 more on naming, comparing and changing angles

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (4)

The angle on the hour
On the hour
$\theta = 30h \;\text{ or }\; 360 - 30h$
The angle at any time
Angle between the hands
$\theta = |30h - 5.5m|$
Above 180°: the smaller angle and the reflex angle
Smaller and reflex angle
$\theta_{\text{small}} = \min(\theta,\ 360 - \theta), \qquad \theta_{\text{reflex}} = 360 - \theta_{\text{small}}$
Naming, comparing and changing angles
Change in the angle
$\Delta\theta = |\theta_2 - \theta_1|$

Watch out for (7)

Count the short way
→ The angle on the hour
The hour hand has moved
→ The angle at any time
Half past is not on the number
→ The angle at any time
Above 180 is the long way round
→ Above 180°: the smaller angle and the reflex angle
Reflex is 360 minus the smaller angle
→ Above 180°: the smaller angle and the reflex angle
Work the angles, do not judge the times
→ Naming, comparing and changing angles
Near 90 still needs the exact value
→ Naming, comparing and changing angles

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