# Definitions of area and volume in $d$-dim spacetime

I am reading Erik Verlinde's paper "Emergent Gravity and the Dark Universe". Equations 2.12 and 2.13 give the area $$A(r)$$ and volume $$V(r)$$ respectively. Do $$d$$ dimensions comprise of both space and time dimensions? Also, $$\Omega_{d-2}$$ is the volume of a $$d-2$$ dimensional unit sphere. Here, are we counting the dimensions of the bulk of the sphere or its boundary. For example, is the regular notion of a sphere a 2-dim or 3-dim object?

Here $$d$$ is the number of dimensions of spacetime, and $$d-2$$ is the number of angular (i.e., non-radial) spatial dimensions. $$\Omega_{d-2}$$ is the integral over those angular dimensions. When $$d=4$$, it is the area of the two-dimensional surface of the usual unit sphere, namely $$4\pi$$.
Here a “$$d-2$$ dimensional unit sphere” means the set $$x_1^2+…+x_{d-1}^2=1$$, not the set $$x_1^2+…+x_{d-1}^2\le 1$$. The “regular notion of a sphere” is a 2-dimensional object, a “2-sphere”, which can be considered the boundary of a 3-dimensional ball. A “3-sphere” is not a 3-dimensional ball but rather the boundary of a 4-dimensional ball.