Starting from a 2D geometry, how can i calculate the volume of the bottle i designed?

Question

i have problems with the bottle producer. whatever i design, even if the proportion are looking allright, the quantity of liquid is wrong. so basically i need, after designing, to verify the quantity of liquid the bottle contains. Thank :-)

Answer 1

It depends on how familiar you are with the Calculus...

If the bottle is "rotationally symmetric" about it's long axis, (like most bottles) you can use a version of the "disk method", to estimate it's volume.

The idea here is to construct an approximate mathematical version of the bottle, by "stacking" together a series of thin, differently sized disks. Since the volume of a disk is easy to compute( V = π r² h,) if you add up the volumes of all the individual disks, you should get a good estimate of the volume of the bottle.

You want to start out by making a series of *equally spaced* measurements of the bottle's diameter, at various points, (perhaps by using a pair of calipers.) I'll call these diameter measurements d1, d2, d3, etc, etc. The volume of the bottle is then given, approximately

V ≈ (π/8) h ([d1 + d2]² + [d2 + d3]² + [d3 + d4]² +.........etc.}

where h is the spacing between the measurements. (Technically speaking, this formulation should be called the "frustum" method, not the "disc" method.) Note that you may want to take into account the thickness of the bottle's walls, then subtract that from your measurements before you do the tabulation, so that you are measuring the *inside* volume of the bottle, not the *outside* volume.

If you are using a CAD program (such as "AutoCAD,") to design the bottle, you can cheat a little bit. Most CAD programs have functions that can calculate both the area and the "centroid" of a two-dimensional figure. From there you can use "Pappus' centroid theorem." to calculate the volume. That is:

V = 2π A Rc

where A is the area of the bottle's two dimensional profile, and Rc is the distance of the centroid from the axis of revolution.

Hope that's not too complicated, please email me from my profile page if you need more explanation....
~W.O.M.B.A.T.