If the radius is infinite
If the radius is >0 and the definition of sphere is used in infinite-dimensional space.
this post was submitted on 25 Dec 2025
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Planes can be thought of as spheres of infinite radius, and have infinite surface area. This point of view is very natural in conformal geometry.
I would assume if and only if the radius is infinite
U'd think, right?!
Math is weird. You can have infinite circumference but finite radius.
I read it as surface area, thus being the amount of space on the sphere itself.
A=4πr2 is the formula if I remember correctly, so I just figure only radius can be altered to match infinity.
Maybe someone will tell me I'm missing something
I believe the surface area of an n-dimensional hypersphere is (n - 1) pi r^{n - 1}. In that case (I may have some factors wrong here, just going off memory), an infinite-dimensional hypersphere has infinite surface area as long as it has non-zero radius.
It does indeed scale with r^(n-1), but your factors are not close at all. It involves the gamma function, which in this case can be expanded into various factorials and also a factor of sqrt(pi) when n is odd. According to Wikipedia, the expression is 2pi^(n/2)r^(n-1)/Gamma(n/2).
Edit: ignore the below, I forgot my pi-factor in the gamma function for half-integers...
Edit 2: Since you're right, my missing gamma-factor completely changes this. An infinite-dimensional hypersphere will have zero surface area for any (finite?) radius.
Original dum-dum:
While I'm completely open that my factor is likely wrong here, the expression you provided is definitely wrong in the 3D case (I'm assuming the r superscript on the pi was a typo), since it doesn't give n = 3 => A = 4 pi r^2.
one can further prove that the sphere S**n−1 can be partitioned into as many pieces as there are real numbers (that is,
pieces)
Would the answer to OP be some argument along the lines of defining the surface area of the ball as the sum of the partitioned balls surface areas then?
IDK what OP is even going on about. This just seemed relevant.
Under the definition of what a circle is...?
It's a polygon with infinite sides. The circumstance would be "being a circle." If it has less than infinite sides, it's not a circle.
Ok but the sum of the infinite sides is a finite value
sphere != circle
Sphere is like a ball (3D), while circles are like those round coasters for drinks (flat)
Okay, and your point is? I did not describe a sphere.
I'm pretty sure the question described a sphere
It said circle originally. OP edited the question after my initial reply.
Fair enough.
Ok but the sum of the lengths of these infinite sides is a finite value
If OP just means the size of the entire inside of the circle, then it would need an infinite radius, too.