How do I arrive at a formula to calculate the volume of the following 3D shape? Does this shape have a proper name?
It kind of looks like an irregular triangular prism with 2 similar triangles as bases. The base edf is bigger in this example, but could be smaller, too.
All edges are different sizes and their lengths are known. Angles α and β are known too. Bases are parallel to each other. ab, df, gf, ga are all right angles.
Couldn't find a formula for a shape like this. I tried subdividing ot into other 3D shapes with known volume formulas, but I wasn't able to infer the sizes of all the necessary dimensions to plug into their volume formulas. Any ideas?
Can / should this be solved using integral calculus or algorithmically, since this can be viewed as the abc triangle riding on the g edge and linearly increasing / decreasing in size until it becomes bfe?..
Best Answer
The solid is a cone (or a pyramid since the base is a polygon) with top cut off. Let $h_1$ be the distance between the planes of the triangles, i.e. the height of the cut pyramid. There's a formula in terms of $h_1$ and $A_1, A_2$ (the areas of the base triangles)
$$V=\frac{1}{3}h_1 \left(A_1 +\sqrt{A_1A_2}+A_2 \right).$$
But you still have to solve the height $h_1$. This can be done by setting the figure into coordinate space by setting the right angle of the bigger triangle to origin and giving the two other points the coordinates $(d,0,0)$ and $(0,f,0)$. Now you want to know the $z$-coordinate of the point that is the right angle point of the smaller triangle. You will know its distance to three points: $g$ to origin and other two can be solved from the diagonals on the sides (they are trapezoids). From there, I think we are able to solve the coordinates.