Hi, I had a conversation with a friend and he brought up this story where one has for example, a square of 1 m2, each side is 1 meter long, resulting in a 4 meter long border. Now, one takes a rigid string that does not stretch at all nor does it shrink, one lays it around the outside of the square and adds 10 cm or 0.1 m to its length, resulting in a 4.10 meter long string.
He now claims that there will be a gap between the 4 meter border of the square and the string of 1.25 cm all around at any point, this also is, according to him, true no matter how long the sides of a square, rectangle or the circumference of a circle are, he says that even if one lays a string around the Equator and then adds 10 cm, this would result in a gap of 1.25 cm all around the Globe between the surface and the string. Can that be true? :dubious:
c = pi*2r
for any 2 circles, the “gap” between the 2 will be equal to the difference between their radii.
r = c / 2*pi, so delta r = (c - c) / 2 * pi
Let’s call pi 3.14 for sake of example
delta r = (c - c) / 6.28
10 / 6.28 = 1.59
So, the idea is there - that the starting size of the body is irrelevant, since the relationship is linear, but the math is off a little because his example applies to rectangles and not circles, which have a different relationship between perimeter and radius than rectangles have betwen perimieter and the length of a side.
eta: I am not a mathemagician.
The story is usually told by adding a yard to a string laid around the earth’s equator. Then the string is six inches above the ground everywhere, because 3 feet divided by 2 pi is about 6 inches.
It’s the factor of 2 pi or 6.28 that’s the heart of this surprising result.
Now if you imagine a sphere with that kind of gap, you’ll get much different results and working through the math will let you understand the perimeter thing better.