Imagine a square with sides of length $10$cm. This square will be fixed: think of it as being glued to the page.

Imagine a second square, of the same size, that slides around the first, always maintaining contact and keeping the same orientation.

In the interactivity below, the second square is red. It has a dot on its top left hand corner.

How far does the dot travel before returning to its starting point?
Try to predict the distance before using the interactivity to check your answer.

 

Change the position of the dot.
How does this affect the distance travelled by the dot?

Change the size of the second square.
What can you now say about the distance travelled by the dot?

Try the same problem with triangles or hexagons instead of squares
(remember your second shape is not allowed to rotate, or overlap with the original shape).

What happens if there are two different shapes?

Is there a theorem here?

Here are two related problems you might like to take a look at:
Rolling Around
Rollin' Rollin' Rollin'