Hello, my name is Murray. I am a grade 10 math student.
My question is:
I think the first issue is that this is not a constant - it depends on the radius of the circle.
I still find it a bit hard to image - but you could ask the same question about an easier space to imagine - the sphere. By a circle, I imagine you mean the set of points equal 'distance' from a fix point. With this in mind, consider the north pole on a sphere, and various circles (cuts through the sphere parallel to the equator). While the cuts themsleves are plane circles, and have a clear circumference, you need to think of the radius as the distance along the sphere from the north pole.
For very small circles, close to the north pole, the answer will be very close to our normal plane ratio. For the equator itself (on a unit sphere) the 'radius' will be pi/2 and the circumference will be 2pi. The value of 'pi' would be 2!
These ratios will vary continuously as you slide the slicing plane down - so the ratio will pass through both rational and irrational numbers, algebraic and transcendental ....
Note that by the time you get the the south pole, the circumference will return to 0 andthe radius will be either pi (if you measure from the North pole) or 0 (if you measure from the other center - the south pole).
On the hyperbolic plane, things will have a similar sphere, except that the values of the ratio will be INCREASING from pi, without limit.
The plane will be the one world in which this ratio is constant.
Walter Whiteley York University