In article ,
W P Dixon wrote:
Ok Guys and Gals,
I do not remember the formula for this to save my life, so I will see if
yall can come up with it. Yes I did check on the web, but did not see the
formula I need.
I want to figure the volume of a gas tank that will not be round or
square, It will have five sides and then the two ends of the tank. With one
end being larger than the other. I would give exact measurements , but being
as I don't know what they will be yet I can't:} I need to find the right
volume in order to get the right measurement . Oh the dilemma !
Be gentle math wizards it's been 25 years since I have had to do this!



It's *not* a single formula.

That said, it's not a difficult problem to solve.

There are at least two approaches that work for your situation:

1) for the 'general case' --
You 'decompose' the object into some simpler ones, calculate the volume
of each of those simpler ones, and 'add em up'.

the 'simpler' forms you want to get to a
A) a cylinder -- top/bottom are parallel to each other, and the same size,
shape doesn't matter, as long both are identical.
B) a wedge -- bottom a parallelogram, tapers to a _line_ at the top
C) a cone -- bottom any shape, tapers to a point at the top.

Then, you simply have to remember one formula for each form:
A) volume of cylinder == area of base * height
B) volume of 'wedge' == (area of base * height)/2
C) volume of 'cone' == (area of base * height)/3


2) for the 'special case' where the two ends are the same shape, differing
only in size, and are parallel to each other --

You can 'extend' the edges that join the ends, until those edges meet.
(they are guaranteed to all come together at a single point.)

You now have a *cone*. You can calculate the volume between the two
"ends" by calculating the volume of the entire cone above the larger
end, and then subtracting the volume of the cone above the smaller end.

the 'height' of the cone, from the 'big end', can be computed by
using _any_ *linear* dimension of the big end, and the corresponding
dimension from the little end, as follows:

To keep things manageable, lets invent some names:
b is a linear dimension of the big end
l is a linear dimension of the little end
d is the distance between the ends, measured perpendicular to the
planes of the ends.
c is the distance that the 'convergence point' is above the big end
A(b) is the _area_ of the big end
A(l) is the _area_ of the little end

c = b/(b-l) * d

so, the volume of the full cone above the big end is:
A(b)*c
and the volume of the full cone above the little end is:
A(l)*(c-d)
thus, obviously, the volume of the 'truncated cone', between the
two ends is:
A(b)*c - A(l)*(c-d)
eliminating the intermediate calculation, by back-substituting for c, gives:
A(b)*(b/(b-l)*d) - A(l)*((b/(b-l)-1)*d)
The area for any given shape is proportional to the square of the
linear dimension of that shape, differences in shape are accounted for
by a difference in the proportionality constant, *only*.

so we get:
k*b^3/(b-l)*d - k*l^2*(b/(b-l)-1)*d
or
k*b*d/(b-l)*(b^2 - (l^2+l-b))

doing it "step-wise" (computing height of cone, and end areas, separately)
is easier grin