Why is it that a square pyramid with the exact same dimensions (length, width and height) as a square prism have a third of its volume? I know this is because the square pyramid formula is the exact same as the square prism except dividing the answer by three at the end, but why do we do this in the first place?
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$\begingroup$Cavalieri's principle is the key.
Take a plane parallel to the given base, at a distance $d$ from the opposite vertex.
The area of the section is $\lambda d^2$, and since $\int_{0}^{1}x^2\,dx = \color{red}{\frac{1}{3}}$, the volume of the pyramid is
$$ \frac{\lambda h^3}{3} = \left(\lambda h^2\right)\cdot\frac{h}{\color{red}{3}},$$ where $h$ is the height relative to the chosen base.
$\endgroup$ $\begingroup$Since this question was migrated here from Mathematica StackExchange and because of the comments by mercio and vonbrand I would like to point out that Mathematica's NIntegrate advanced documentation for Duffy coordinates has a related theorem and an illustrating plot.
Here is the theorem:
Theorem: A $d$-dimensional cube can be divided into $d$ disjoint geometrically equivalent $d$-dimensional pyramids (with bases $(d-1)$- dimensional cubes) and with coinciding apexes.
Here is the illustrating plot:
And here is Mathematica code to make the plot:
grx = GraphicsComplex[{{0, 0, 0}, {1, 0, 0}, {1, 0, 1}, {1, 1, 1}, {1, 1, 0}}, {Polygon[{1, 2, 3}], Polygon[{1, 3, 4}], Polygon[{1, 4, 5}], Polygon[{1, 5, 2}], Polygon[{2, 3, 4, 5}]}];
gry = MapAt[Map[RotateLeft[#] &, #] &, grx, {1}];
grz = MapAt[Map[RotateRight[#] &, #] &, grx, {1}];
Graphics3D[{Opacity[0.5], Red, grx, Cyan, gry, Yellow, grz}]I did this in high-school and failed to appreciate the importance and subtlety of this result that was a big discovery in Antiquity. This question of yours determined me to try to understand the classical proofs. I think I managed that with the material at these two links: ( some nice picture but did not understand the argument as to why some pyramids have the same volume) and this one, where they have a nice picture of a pyramid with two colors that finished the proof for me. I suggest looking at these and other materials.
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