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This is probably a trivial problem but I need some help with it anyway. Lets say we have two planes that consist of four points each in a three dimensional space. How can I calculate the volume between these two planes?

To make the example more concrete let's say that the planes points (x,y,z) are:

Plane 1: (1,0,0), (2,2,0), (3,0,0), (4,2,0)
Plane 2: (1,0,2), (2,2,2), (3,0,2), (4,2,2)

Thanks!

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You mean the prism formed by those two polygons? Just figure the area of one of those polygons and multiply by the height of your prism. –  J. M. Oct 7 '11 at 9:39
    
Hmm... I'm not familiar with prism but after your description it looks like x and y have to be the same? since you only multiply by by the height (z)? What if plane 2 looks like: (3,0,2), (2,6,2), (1,1,2), (4,2,2) Can I still use that method? –  picknick Oct 7 '11 at 9:49
    
In that case, what then is the volume you speak of? –  J. M. Oct 7 '11 at 9:54
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Maybe you should post a picture to help us see what you want to happen... –  J. M. Oct 7 '11 at 9:56
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2 Answers

In fact you have to calculate the volume of prism which has for base parallelogram (see picture bellow) .So we may write next expression:

$V=BH$, where $B$ is area of the base and $H$ is height of the prism which is equal to the value of z coordinate, $2$.

Since base is parallelogram we may write $B=ah_B$ ,where $h_B$ is height of the base,and $a$ is length of side of the parallelogram

According to picture bellow $h_B=2$ and $a=2$ also,so

$B=ah_B=4$, Now we can find $V$ as:

$V=BH=8$

enter image description here

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Your example has two features that make it easy:

  1. Plane 1 and Plane 2 are parallel. Will this always be the case?
  2. The lower polygon is simply a projection of the upper polygon. According to your comment, this will not always be the case.

If the second condition is not satisfied, then the two polygons don't unambiguously define a region of 3-dimensional space. For example, if the two polygons are ABCD and EFGH, do you mean the skew polytope with edges AE, BF, CG, and DH? But then the face AEFB is not flat, and you have to specify its surface in some way.

So you must decide precisely what you want your volume to contain. Then we can help!

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