Area of 2D fractal?

Question

Some fractals have a whole fractal dimension, can their measure be calculated? For example if you start with a tetrahedron of a given size and recursively remove the central octahedron leaving 4 tetrahedrons of half the side length you will end up with a fractal of dimension 2. How does the area of this fractal relate to the size of the original tetrahedron? There is an image of this fractal here. What if you start with a cube, divide it into eight smaller cubes, remove the four cubes of a given parity and continue recursively as shown here? If you generate the Type 2 Quadratic curve (dimension 1.5) on one plane and a 0.5 dimensional Generalized Cantor Set on the perpendicular axis, produce a plane going through every point in the Cantor set and project the curve on each plane, what is the area of this?