Math Circle experiment

Welcome to math help online,

The circumference
divided by the diameter is the same for every circle' is only true if
you're talking about a 'flat' space. But here's an experiment you can
do to get a feel for what it would be like to live in a 'curved'
space:

Look at a globe, like this one:

    

Call the horizontal circle that goes through the top of South America
(i.e., the equator) 'Circle 1', and the horizontal circle that goes
through Chicago 'Circle 2'. more examples on online math tutors;
If you measure the 'radius' of each circle
along the surface of the earth from the north pole (which is the
'center' of each circle), then dividing circumference by diameter
gives you two different values of pi, doesn't it?

In fact, if the earth were perfectly spherical, the 'radius' of the
equator (measured along the surface) would be 1/4 of its circumference
(do you see why?), so the ratio of circumference to diameter - that
is, 'the value of pi' - would be exactly equal to 2, rather than
3.14...
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