Thisactivity is about Analog clocks and the angles made by thehands of the clock. You can find out more about angles and how they"re measured on thepage Degrees(Angles).

You are watching: 1:30 clock

### What is the angle between the hands of a clock at 1 o"clock? At 1 o"clock the minute hand(red) points to the 12 and thehour hand (blue) points to the 1.So we need to find the angle betweenthe 12and the 1.

### How many of this angle are there in a complete turn? There are 12 of them in a complete turn (360°), so eachonemust be360°÷ 12= 30°

So the angle betweenthehandsof a clock at 1 o"clock is30°.

Note:

It doesn"tmatter whether we are talking about 1 am or 1pm, the answer is exactly the same for both.The angle between the hands at1 o"clock could also be given as the reflex angle 330°,but we willalways give the smaller (acute or obtuse) angle.

### What is the anglebetween the hands of a clock at 2:30? At 2:30 the minute hand(red) points to the 6 and the hour hand(blue) points halfway between the 2 and the 3.

### So how many lots of 30° do we have this time? The angle between the 5 and the 6 is30°The angle between the 4 and the5 is30°The angle between the 3 and the4 is30°The remaining angle is½ × 30° = 15°

So the anglebetween the hands of a clock at2:30 = 30° + 30° + 30° + 15° = 105°

### Complete the following table (give the smaller angle in each case):

 Time 1:00 2:30 7:00 10:30 11:20 3:40 5:15 8:45 Angle 30° 105°

## More Complicated Times

Finding the angle between the hands of a clock is easy as longas we don"t use complicated times.

For example finding the anglebetween the hands at 9:37 is much more difficult. You can trythat one if you wish, but it"s probably too hard.

### Example: Atwhat times is the angle between the hands of a clock equal to30°?

Notice that the question asks for "times".There are many possibleanswers.Some of them are easy to find, others much more difficult.  1 o"clock 11 o"clock But what aboutthis one? 4:15 PM

At first glance, it looks like this might be a 30° angle also,but at 4:15, the hour hand has already moveda quarter of the way between the 4 and the 5.

So theangle is30°+ ¼ × 30° = 30° + 7½° = 37½°.

This might be a more accurate answer: Can you find more 30° angles like this one?

## At what times of the day do the hands of a clock lie in a straightline?

In other words the angle between them is 180°?

One obvious answer is 6 o"clock: ### But what other answers could there be?

9:15 is not correct for a similar reason that 4:15 didn"t give us exactly a30° angle ... the hour hand has moved on beyond the 9.

See more: Which Of The Following Items Would Not Appear In An Income Statement?

This seems to be a very difficult question to answer, but there is aneasy way.

### How many times between 6:00 am and 6:00 pm do the hands make astraight line?

There must be at least one point time for each hour:

one between 7:00 am and 8:00 am, one between 8:00 am and 9:00 am, one between9:00 am and 10:00 am, etc, approximately ...one between 4:00 pm and 5:00 pm

That makes 11 equal parts, and so: