Of course daniel is right that the three-dimensional volume of the part of a sphere in $\mathbb R^3$ between two parallel planes at a distance $d$ is less near the poles. I suspect that what you meant to claim is that the surface area of the sphere between two parallel planes depends only on their distance $d$.
This is because the surface area of an infinitesimal ring between two parallel planes at an infinitesimal distance $\mathrm dx$ at a distance $x$ from the centre of a sphere of radius $r$ is given by the perimeter $2\pi\sqrt{r^2-x^2}$ of the ring times the infinitesimal width $\mathrm dxr/\sqrt{r^2-x^2}$ of the ring, and the product $2\pi r\mathrm dx$ is independent of $x$.
In $n$ dimensions, the perimeter is replaced by an $(n-2)$-dimensional volume proportional to $(r^2-x^2)^{(n-2)/2}$, whereas the width remains the same, so the product is proportional to $(r^2-x^2)^{(n-3)/2}$. This is independent of $x$ only for $n=3$.