I am reading an introductory book on Fractals where the following intuition is given for Hausdorff dimension of a smooth curve: if we define "length" as $L(r)=N(r)\cdot r$, with $N(r)$ straight line segments of length $r$ that step around the curve. As $r$ approaches $0$, $L(r)$ approaches the length of the curve.
For finite-length smooth curves, this approaches a finite length $L$. However, for fractals embedded in a one-dimensional space, this is not so, so we can define a "dimension" $D_H$ such that $N(r)\cdot r^{D_H}$ remains finite.
This means we can define
$$D_H:=\lim_{r\to0} \frac{\log(N_r)}{\log(1/r)}.$$
I understand how this intuition holds for fractals embedded in one dimensional space. However, my issue arises when the book attempts to extend the induction to higher dimensions. For example, the 3-D Sierpinski gasket: start with an equilateral tetrahedron and remove the open central upside down equilateral tetrahedron with half the side length. Repeating the process with $n$ iterations, there will be $3^n$ tetrahedron of side length $r=r_0({1/2^n})$. The book then goes on to say that the Hausdorff dimension can be thought of as
$$D_H=\lim_{n\to\infty}\frac{\log(4^n)}{\log(2^n)}=2.$$
However, wouldn't the "measure" of each individual tetrahedron be more appropriately expressed as the volume of said tetrahedron: changing proportionally to $1/2^{3n}$ with $n$ iterations? Wouldn't this make
$$D_H=\lim_{n\to\infty}\frac{\log(4^n)}{\log(2^{3n})}<1?$$
I know this is by no means the formal definition of Hausdorff dimension but I do not have enough background to understand the formal definition. Is there an issue simply with extending the intuition, or with my using the area of triangles instead of lengths of their sides?
Edit: Changed triangle to tetrahedron to make the limiting procedure easier.
Area is a two-dimensional concept. The basic idea is that $d$-dimensional measure scales with the $d$'th power of length. So if you have an object that scales so that magnifying lengths by a factor of $r$ gives you $r^d$ copies of the original, that object should have dimension $d$.
EDIT: For example, consider your 3-d Sierpinski gasket. If you magnify the original by a factor of $2$, you get four (not $3$) objects the same size as the original. Since $4 = 2^2$, the gasket has dimension $2$.