We have a circular cross section here. The area is proportional to the square of the radius, so the radius is proportional to the square root of the area. If the area increases from 1 area to 3.4 areas (a 240% increase), then the radius should increase from 1 radius to 3.4^0.5, or about to 1.84 radii.
Note that it doesn't actually matter that the cross section is circular. Imagine that we have some arbitary cross section and then draw a circle around that arbitrary cross section. The cross section is some fixed proportion of the area of the circle. If we scale both the circle and the cross section (about any point!) then that proportion doesn't change. Doubling the size of the circle is also doubling the size of the circumscribed shape. No matter the cross section, scaling the verts defining it by x will increase the cross section by x^2.
If we have clean topology, we can select all of the circumferential loops in edge mode and simply scale about individual origins by 1.84:
If we don't have clean topology, and we have some kind of horrible point cloud, we can make a temporary mesh deformer that has decent topology and use that to scale the mesh instead:
Here, of course, we're exactly increasing the area of the mesh deformer's cross-sections, which isn't the same thing as exactly increasing the area of the deformed mesh's cross-sections. However, as we use more and more geometry on the mesh deformer to more exactly approximate the original mesh, we approach perfect agreement.
You could make a deformer out of knife cuts from your original mesh, in which case you'd have perfect agreement: