Geometry - Volume of a distorted tent

Question

How would one calculate the volume of a tent shaped object with the upper edge not parallel with the base plane of the tent?

edit: The tent has a rectangular base with two poles at different heights at the edges of the base.

Answer 1

This is for a tent which has a rectangular base $a \times b$ and with the center pole holding up the tent having heights $h_1$ and $h_2$ at the two ends which are at the centers of the two rectangle sides labeled $a$. The height at a point $x$ units from the $h_1$ end is (using the coordinate $0$ at that pole and then $x$ going along the midline of the rectangle) $h(x)=h_1+[(h_2-h_1)/b]\cdot x.$ Then the cross-sectional area at point $x$ is $(1/2)a\cdot h(x),$ which when integrated from $0$ to $b$ and simplified gives the volume formula $$V=\frac{ab(h_1+h_2)}{4}.$$ Note that when $h_1=h_2=h$ this gives the right thing.

It would be more complicated if say the pole holding the tent up was not only at different heights at the two ends, but also askew with respect to the rectangular base.