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I'm tryin to calculate the volume of a three dimensional shape which, from all three angles, looks a bit like a staircase. A way to visualize it: it's like a bunch of boxes all piled up against the corner of a room.

I made an example of one side in paint (sorry), all three sides will look like this,

see image.

What I know:

  • The surface area of all three sides
  • The three dimensional coordinates of each corner
  • The end point of each axis (in this example x=5 and y=4).

The numbers are corresponding to the length of the wall, so this wall has a width of 4 and a height of 5, which makes the surface area of the entire wall 20. Lets say the surface area (or the area under the straircase) is 12. And lets say that the other surface areas (or areas under the staircases are 10 and 14).

Is there a way to calculate the total volume under the three dimensional staircase knowing all three surface areas or any of the other properties?

Thanks!

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  • $\begingroup$ Is that supposed to be $1+2+3+4$ so the side we can see has area $10$? $\endgroup$
    – Henry
    Commented Mar 11, 2021 at 18:18
  • $\begingroup$ No sorry, the sides are corresponding to the length of the wall. So this wall has a width of 5 and a height of 4, the surface of the entire wall would be 5*4=20. I'll edit it, thanks. $\endgroup$ Commented Mar 11, 2021 at 18:25
  • $\begingroup$ Lets say we have 50 boxes. We pile 30 of those 50 boxes up against both vertical planes/walls, if we look at the room from above we would see sort of an L-shape of piled up boxes, right? Now we can allocate the remaining 20 boxes inside that L-shape, also creating a staircase shape on the floor of the room (as seen from above). I will try to make a 3d image if it's still unclear. $\endgroup$ Commented Mar 11, 2021 at 18:55
  • $\begingroup$ If both the vertical planes are $5\times4$ then the view from above will be $5\times5$ so it can't look exactly the same. $\endgroup$
    – nickgard
    Commented Mar 11, 2021 at 19:07
  • $\begingroup$ That's true, that's why the surface areas are different from each other. $\endgroup$ Commented Mar 11, 2021 at 19:25

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From your explanation, I gather that the staircase shape is actually a box-wall and not just a silhouette casted, but it's actually impossible to have all 3 to be the same shape unless they have the same endpoints (x=y=z=5 for example). Even then, the 3D could have taken many actual shapes within the minimum and maximum volume. You can imagine cutting off a corner of a solid cube so that the sliced surface is an equilateral triangle, and you can see that all the other faces of the pyramid are 3 exactly identical isosceles right triangles. This is analogous to your conditions, but you can also scoop out the filling inside this shape from the equilateral triangle face (the scooped out shape is the same as original but smaller) so that the 3 identical faces are left intact but this time your shape is hollow, just 2 walls and a floor sharing a vertex and 3 edges. Therefore, for your original "Is there a way to calculate the total volume under the three dimensional staircase knowing all three surface areas or any of the other properties?", the answer is "no" but you can find the range of its volume.

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