Imagine, you have a shape which is almost spherical and you want to find the volume. Now, the shape is N-dimensional so you are considering it as N-sphere. End goal is to find the volume as precisely as you can by approximating it as a sphere.
Now, I have distances from "center" (center is a point which best give a spherical shape) to "surface" for different surface points. I can call them my radii. These radii will be equal for a sphere but nearly equal for my shape. Now, to find the average volume, do I take averages of these radii first and then put it in my N-sphere volume formula or I put every radii in my N-sphere volume formula and then take the average of them? Intuitively, these two type of averages are different for any sphere (D!=2).
Let's take an exxagerrated example, I have american football shaped object whose antipodal points are measured. You can find the radius from that. The radii near minor axis is alot smaller than near major axis'. So, route for finding the averages is a question one should ask before directly calculating the volume.