We had several great solutions for the first part of this problem from pupils at Highcliffe Primary School. Whitney and Joe said:

The new shape (i.e. the locus of the centre of the circle) will be a square with rounded corners.

Sam and John explain that the length of the locus ...:

... is the perimeter of the old square plus that of the circle.

Can you remember the special name we give to the perimeter of a circle?
Unfortunately nobody has sent in solutions to the second part of the question which asks about a circle travelling round an equilateral triangle. Please send in your ideas!