Fun fact: the circumference of a circle in our spatially flat Universe is 2πr. But if our Universe was positively curved (think the surface of a sphere), it's *less than* 2πr. If negatively curved, like a saddle or pringles chip for example, it's *greater than* 2πr!
It makes sense too if you picture it: on a sphere, the circumference of a circle of the same radius as a circle on flat space will be smaller because the sphere curves inward, making the path you travel that circle to be shorter than the one on a flat sheet of paper.
Now do the same on a pringles chip: the path you travel along the circumference is *longer*. Imagine a sheet that's kinda wavy around a point, and you draw a circle around that point. that's *kinda* like negatively curved spacetime, so a circle would have a larger circumference.
Cool thing is you can literally derive the circumference for flat, positively, or negatively curved spacetime, all from this equation called the metric! I'll do this for you all after the class I teach so you can see the awesomeness for yourselves.
Okay! it begins with the metric, where dl is the path between two points, dr is how much we move *radially* (ie, from center outwards), S_κ gives us the dependence based on the curvature of the Universe, and dΩ tells us how the angles change and which ones do: 
Next, we need to figure out what's constant: the radius of a circle doesn't change, so dr goes to zero. There's also an angle that's fixed, and we fix θ to π/2 since we integrate in spherical coordinates, and it's over φ that we integrate right around to 2π
(θ only goes to π so we don't double count!)
So we fix θ to π/2 and set the circumference to be on the "equator". And since θ is fixed, then it's not changing, so dθ = 0.
So we fix θ to π/2 and set the circumference to be on the "equator". And since θ is fixed, then it's not changing, so dθ = 0.
Now we have the metric down to something simple to work with! So let's set up the equation for flat space (ie, κ = 0):
Finally we integrate and TADA WE HAVE THE CIRCUMFERENCE OF A CIRCLE IN FLAT SPACE!
Then we can find it for κ=1 (positive) and κ=-1 (negative) curved spacetime. Notice the R_0: that's the radius of curvature. Ours is flat so there's no R_0 dependence. Next, note that sin θ is always less than θ, hence C < 2πr. But sinh θ is greater than θ, so C > 2πr.
And that concludes today's fun with circles and their circumference in curved or flat spacetime! 🙂
