# Within a space which is a 3-sphere, what the is surface area of a 2-sphere?

If an ant lives on a ball (2-sphere) with radius $$r$$, draws a circle and measures the radius of the circle as $$R$$ as it walks along the ball's curved area, it will find that the length of the circle is $$U = 2\pi r \sin (R/r)$$.

Question: raising all dimensions by one, what is the area of a 2-sphere if it lives in a 3-sphere? By raising all dimensions by one, I mean

• space to live in: 2-sphere -> 3-sphere
• geometrical object created: 1-sphere (circle) -> 2-sphere (ball)
• length of circle $$U$$ -> area of 2-sphere $$A$$
• measured radius of of the object created: $$R$$ in both cases
• radius of space we live in: $$r$$ in both cases

What is $$f$$ in $$A = f(R, r)$$?

An $$n$$-sphere of radius $$R$$ in an $$(n+1)$$-spherical space of radius $$r$$ (with $$R$$ measured inside the spherical space and $$r$$ measured in $$\Bbb R^{n+2}$$ where the $$(n+1)$$-sphere is defined) is an $$n$$-sphere of radius $$r\sin\frac Rr$$ in a flat (i.e. Euclidean) $$(n+1)$$-dimensional cross section of $$\Bbb R^{n+2}$$. (This is easiest to visualize with $$n=0$$.) The surface measure is the same in both spaces (unlike the interior volume!), so we can plug $$r\sin\frac Rr$$ into the appropriate standard formula to compute it. For $$n=2$$ this gives $$4\pi (r\sin\frac Rr)^2$$.