For a generic base convex polygon, does the same argument apply? Is the volume equal to A/n sum_{i=1}^{n} h_i, where n is the number of vertices of the base/top?

Will Underwood 2022-09-24 Answered
Volume of a Truncated Right Prism with generic base convex polygon
I can find lots of pages online saying the volume of a Truncated Right Triangular Prism is A 3 ( h 1 + h 2 + h 3 ), where A is the area of the base and h i is the height of vertex i of the top face. Or, in other words, to find the volume you "flatten" the top face by finding the average height of its 3 vertices, and then the volume is just that of a regular prism.
My question is: for a generic base convex polygon, does the same argument apply? Is the volume equal to A n i = 1 n h i , where n is the number of vertices of the base/top? Is there a proof for this anywhere?
My intuition says it's true, but what if the base shape was like a really long kite; wouldn't the height of the vertex at the bottom tip of the kite skew the average height of the top face?
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Answers (1)

skarvama
Answered 2022-09-25 Author has 5 answers
Step 1
Your formula for the volume cannot be true, in general, for n > 3. Here's a counter-example with n = 4.
Take a truncated right quadrangular prism with its base on the z = 0 plane and its top vertices given by:
V 1 = ( 0 , 0 , 0 ) , V 2 = ( 1 , 0 , a ) , V 3 = ( 0 , 1 , b ) , V 4 = ( 2 , 2 , 2 a + 2 b ) ,
with a and b positive constants. Note that those points all lie in the same plane, because V 1 V 4 = 2 V 1 V 2 + 2 V 1 V 3 .
Step 2
The volume of this solid can be computed dividing it into two truncated triangular prisms with a plane passing through y-axis and V 4 . Both their bases have unit area, hence applying the formula for the triangular case with h 1 = 0, h 2 = a, h 3 = b, h 4 = 2 a + 2 b we get:
V = A 1 3 ( h 1 + h 2 + h 4 ) + A 2 3 ( h 1 + h 3 + h 4 ) = 1 3 ( 3 a + 2 b ) + 1 3 ( 2 a + 3 b ) = 5 3 ( a + b ) .
On the other hand, if your generalised formula were true, we would have:
V = A 1 + A 2 4 ( h 1 + h 2 + h 3 + h 4 ) = 3 2 ( a + b ) .
Hence the generalised formula doesn't work.
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