Watching the Wheels Go 'round and 'round

Age 7 to 11
Challenge Level

Seven pupils of Wrenbury Primary School's Extension Maths Club had the following solutions:

First you need to know how many centimetres in a metre ($100$).
Then you take $200$cm and convert them into metres, which gives you $2$ metres.
Therefore, the circumference is $2$ metres.
Then you need to know how many metres in a kilometre ($1000$).
So, you halve it to find out how many times the wheel turns ($500$ times).
Then you do the same calculations for the back wheel, which is $50$cm in circumference or $0.5$m.
This means that the back wheel would turn $4$ times more than the front wheel ($2000$ times).
Therefore, the back wheel must get more wear and tear because it goes round more times than the front wheel does.
Also, half of the circumference of the wheel is greater than the diameter of the tyre.

Ah, but how do we know that?

Well, Daniel suggests using good sense and logic to figure out that half the circumference is bigger than the diameter:

Just look at a circle and you can see.

There is agreement with Daniel from Camilla, Phillippa, Hannah and Laura, all from The Mount School:

The diameter is a straight line.

What we do know is that the shortest distance between two points is a straight line. So, the arc of the circle or curve of the circumference will make it longer than the straight line of the diameter.

Now, why is it that half of the circumference is greater than the diameter, or the diameter is less than half of the circumference?

My guess is that you want to know if the diameter is more or less than the distance around half of the circumference ...

... said Catherine wisely. She went on to give a wonderful mathematical explanation of her thinking:

The equation for the circumference is $C$ (circumference) $= 2 \times \pi \times R$ (radius)
This means that in the case of the bigger tyre, the circumference is $2$m.
Therefore, half of the circumference is $1$m.
This means that $2$m $= 2 \times \pi \times R$ and the radius works out to be $0.318$.
Because the diameter is twice the radius, the diameter would be $0.636$m.
The diameter is less than the $1$ metre distance around half of the circumference.
With the smaller tyre, with a circumference of $0.5$ metre, the radius works out to be $0.08$m, and diameter is $0.16$ metres. Again, this is smaller than half the circumference.

Christine says:

I think that the diameter will always be less than half of the circumference.

Do you agree with her?

The answers above were supported by Jason and Matthew from Tattingstone School and Crewe, Errington, Porter, and Croft as well as Tom ("the Tornado"), who wrote how he "is enjoying this - I love maths". Good for you Tom!