I'm looking for a formula to describe surface and volume of a sphere in hyperbolic 3-space. I found some results which were generalized for any dimension, but I wasn't able to understand them.
Overall, I find there's a sort of "symmetry" between spherical and hyperbolical geometry in these formulas. The circumference of circle in hyperbolic geometry is $2\pi\sinh(r)$, in spherical geometry it's $2\pi\sin(r)$. Areas are $2\pi(\cosh(r) - 1)$ and $2\pi(1 - \cos(r))$, respectively.
I then tried to generalize it into 3D. I was trying to find surface and volume of sphere in spherical space (considered as a cap of Euclidean 4D hypersphere), and eventually obtained formulas that seem to work: $2\pi(1 - \cos(2r))$ for surface and $2\pi(r - \sin(2r)/2)$ for volume. Considering the analogy between formula for area of circle and for surface of sphere, the hyperbolic formulas should be $2\pi(\cosh(2r) - 1)$ for surface and $2\pi(\sinh(2r)/2 - r)$ for volume, but this is based mostly on intuition, so I'd appreciate if you could either confirm my line of thought or give me the real result :)
(Incidentally, what is the formula for length of an equidistant curve? I used the same analogy between goniometric and hyperbolic functions to solve it for sphere: if I have a certain length a of straight line, I can construct the perpendicular lines at the endpoints which will then intersect the equidistant curve (circle on sphere, hypercycle in hyperbolic geometry) under right angles as well. If b is the distance of the equidistant from the line, the length of such defined arc of equidistant is acos(b) on sphere, so it should be acosh(b) in hyperbolic geometry -- is this correct? Finally, what would be the formula for a part of surface equidistant from plane directly "above" a part of the plane of known volume?)