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Rolling Around

Age 11 to 14 Challenge Level:

A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square.
Describe the locus of the centre of the circle and its length.

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If the circle now rolls around an equilateral triangle, can you describe the locus of the centre of the circle and its length?

Can you generalise your findings?

Here are two related problems you might like to take a look at:
Rollin' Rollin' Rollin'
Is There a Theorem?