Clock angles

Hello, and welcome to Myriora Maths! At half past three, what is the angle between the two hands of a clock? Most people say ninety degrees. Most people are wrong!

A full turn is three hundred and sixty degrees. There are twelve numbers on a clock, so there are twelve gaps. Each gap is three hundred and sixty divided by twelve: thirty degrees.

The minute hand goes all the way round in sixty minutes. So it turns six degrees every minute. At thirty minutes past, it has turned thirty times six: one hundred and eighty degrees. It points at the six.

Now the hour hand. At three o'clock, it points exactly at the three: ninety degrees from the twelve. But in the next thirty minutes, it moves too! Half an hour moves it half a gap: fifteen degrees. So at half past three, it's at ninety plus fifteen: one hundred and five degrees.

So the angle between the hands is one hundred and eighty, minus one hundred and five. That's seventy-five degrees, not ninety!

Here's the quick rule. The minute hand turns six degrees a minute. The hour hand turns thirty degrees an hour, which is half a degree a minute. Work out where each hand is, measured from the twelve, then find the difference.

A common mistake is forgetting that the hour hand moves. At half past three, it isn't on the three any more. So the angle isn't three whole gaps, ninety degrees. It's two and a half gaps: seventy-five degrees.

Your turn! What is the angle between the hands at twenty past two? You've got twenty seconds, or pause the video if you need more time. Go!

Time's up! The minute hand has turned twenty times six: one hundred and twenty degrees. The hour hand starts at two o'clock, sixty degrees. In twenty minutes it turns twenty halves: ten more, so it's at seventy degrees. One hundred and twenty minus seventy is fifty degrees!

Quick review. Each gap on a clock is thirty degrees. The minute hand turns six degrees a minute. And the hour hand turns half a degree a minute, so don't forget it moves too! See you next time!