The cuboid volume formula and the dissection of a cuboid into a cube

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.cuboiddissection0

Okay so we say that we have a cuboid with dimensions a \times b \times c where we conventionally say that it has volume abc meaning it has the same volume as a cube with side \sqrt[3]{abc}. 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 a \times b face and dissect it into a rectangle having one side \sqrt[3]{abc} and the other (by the area formula) being ab / \sqrt[3]{abc}.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 \sqrt[3]{abc} as one of it’s side.

Note the depth c of the cuboid is unaffected by this procedure.

cuboiddissection0

(You might need more pieces than the image indicates)

Next we turn out attention to the face with sides ab/\sqrt[3]{abc} and c and perform the same procedure where this face is transformed into a rectangle with one side being \sqrt[3]{abc} by construction and the other side

\cfrac{\text{Area of face}}{\sqrt[3]{abc}}= \cfrac{abc / \sqrt[3]{abc}}{\sqrt[3]{abc}}=\sqrt[3]{abc}

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 Bh 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.

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