# Surface area of a d-sphere

I'm trying to follow Schwabl Thermodynamics, and I found the following equation for the surface area of a unit d-sphere:
$$\int d\Omega_d = \frac{2 \pi^{d/2}}{\Gamma(d/2)}$$

But this formula clearly fails for d=1:
should be $$\pi$$
and d=2:
should be $$4 \pi$$. What gives?

It is the difference between a "sphere" and a "ball". A "2-ball" is the a two dimensional disk, which might have equation $$x^2+ y^2\le r^2$$, while the "2-sphere" is the surface of a "3-ball" and might have equation $$x^2+ y^2+ z^2= r^2$$