I'm a bit stuck on a geometric problem and after searching on Google, cannot find anything suitable – partly because I don't really know what to search for.
I've got a number of cross sections from along a river channel and want to work out a generalised volume for the channel. The cross sections are all separated by known distances, but the sizes and shapes vary.
I am not very au fait with maths, but I believe the Cavalieri principle can be applied when the shapes vary but the area remains the same?.. or have I got this wrong and can it be used or adapted to work for this problem?
I'd be greatful if anybody could help me with a formula or point me in the right direction of where to look?
The image below illustrates what I'm talking about.
Thanks
Cobain
Best Answer
Notice that in the diagram, three of the four vertices on the "river bottom" at A are connected to the same vertex at the left side of the "bottom" of B. If you were to connect the vertices differently, for example connect three of the vertices of A to the "right bottom" vertex of B, you would form a solid with a larger volume.
Given that you do not really know how the actual cross-section of the river varies between the measured cross-sections, you have to take a guess. You could linearly interpolate the cross-sectional area, but if you apply that method to several cross-sections parallel to the base of a pyramid, you don't get back the volume of the pyramid. An alternative might be to use the formula for the volume of a frustum:
$$ V = \frac13 h(A + B + \sqrt{AB})$$
where $A$ and $B$ are the areas of the two bases of the frustum (corresponding to the two cross-sections A and B in your figure) and $h$ is the distance between cross-sections.