Rolling Around
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.
Bill from Dana Middle School has given us a more detailed
description of the path of the centre of the circle:
As the circle rolls along the
outside of the square, its centre traces a path one radius distant
from the side of the box, parallel to the side, and the same length
as the side, until the circle gets to a corner. At the corner, the
point on the circle stays fixed to the corner, while the centre of
the circle traces a circular path of radius r around the corner,
until the radius (from the corner of the square, now) has swept out
a certain angle (in this case, 90 degrees). The sweep begins with
the radius at a right angle to the side the circle is leaving, and
ends when the radius is at a right angle to the side it is
entering. Since it does this four times, the length of the path
around the corners of the square equals the circumference of the
circle.
In effect, in circumnavigating the
corners, the centre of the circle rotates completely around a point
on the circle's circumference one time, and, since the distance
from the centre to the edge is the same as the distance from the
edge to the centre, this part of the path is the length of the
circumference. On the straightaways, the length of the path is
equal to the length of the sides, the total is the perimeter of the
square.
The length of the path is the
perimeter of the square plus the circumference of the circle.
By like reasoning, when the circle
rolls around a triangle, as it rounds the corners the circle will
turn around completely once, which results in the centre of the
circle tracing out the circumference, which, when added to the
perimeter of the triangle will give the length of the path.
In general, as a circle rolls
around a convex polygon, the length of the path of the centre of
the circle will be the perimeter of the polygon plus the
circumference of the circle, the locus of the path being a distance
away from the polygon equal to the radius of the circle.
A very thorough explanation, thank you Bill.