So at the end of the last post I gave the outline of the idea of how you might extend the dissection of a rectangle into a square and the definition of area to an analogous procedure for the cuboid and it’s volume. The idea is really simple but I thought I’d practice my diagram writing skill by providing an illustration and expand the definition of volume.
Okay so we say that we have a cuboid with dimensions where we conventionally say that it has volume meaning it has the same volume as a cube with side . Let us use dissection to motivate this; that is describe a procedure of cutting up the cuboid and reassembling it into the prescribed cube.
We will essentially perform regular rectangle to square dissections face-wise splitting the cuboid into prism.
Let us start with the face and dissect it into a rectangle having one side and the other (by the area formula) being .Applying this to the whole cube the cuts through the face are extended through the solid along planes perpendicular to the face forming prism which are rearranged to form a new cuboid with as one of it’s side.
Note the depth of the cuboid is unaffected by this procedure.
(You might need more pieces than the image indicates)
Next we turn out attention to the face with sides and and perform the same procedure where this face is transformed into a rectangle with one side being by construction and the other side
And so this face is square! All the sides of this cuboid are therefore equal and we’ve ended up with a cube.
*This procedure is not original but my interpretation hasn’t been checked so some caution should be taken
Sidenote: Just as triangles and polygons can be dissected to rectangles a prism can therefore be disected into a cube as well and paying attention to the details of such a procedure you recover the conventional formula for general prism. For the cylinder we don’t have any elementary means of dissecting into a cube but just as you may approximate a cirvle by triangles you can approximate the cylinder by triangular prisms and obtain the standard formula.