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Most of the informal proofs about volume of pyramid I have seen involves cutting cube into 3 like this:

Pyramid inside cube

Then it skips to statement that volume of any pyramid is $\frac{bh}{3}$ where $b$ is base and $h$ is height.

How does cutting a cube into 3 pieces proves that it is true for any pyramid(let's say star pyramid, rectangular pyramid,...) is $\frac{bh}{3}$?

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    $\begingroup$ I highly recommend you check out Cavalieri's principle en.wikipedia.org/wiki/Cavalieri%27s_principle $\endgroup$
    – Oria Gruber
    Jan 23 at 5:04
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    $\begingroup$ An informal proof: P is the center of the cube. The volume of a cube of edge a is $a^3$. We can see 6 equivalent pyramids, each with a vertex at P and having one of the cube faces as a base. $V_{1pyr}=α.(a^2).(\frac{a}{2})$ $a^3=6.α(a^2)(\frac{a}{2})$. So $α=\frac{1}{3}$ $\endgroup$
    – Wilson Massaro
    Jan 26 at 21:27

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Suppose you know that any pyramid with a square base has volume $\frac13 bh$.

Next, suppose the base (with area $b$) is a shape made up of $k$ squares (with areas $b_1, \dots, b_k$). We can chop up the pyramid into $k$ pyramids with square bases, which have volumes $\frac13b_1h, \dots, \frac13b_kh$. The total volume will be $\frac13(b_1 + \dots + b_k)h$, or $\frac13bh$.

Next, suppose the base is any other shape. (The pyramid could be a cone, a pentagonal pyramid, whatever.) We can approximate the base arbitrarily well with shapes made up of many tiny squares (that's how your computer screen works). Since all those approximations have volume $\frac13bh$, so does the real pyramid.

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    $\begingroup$ I saw that Jacob Claassen's answer already addressed the issue of height, so I did not. $\endgroup$
    – Misha Lavrov
    Jan 23 at 6:20
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Consider what happens when you elongate any 3d shape along one axis by a factor of $n$: the volume is multiplied by $n$. Now consider what happens when you move the tip of the pyramid so that $h$ remains constant: all cross sections of the pyramid remain the same, so the area is the same. It follows that if one pyramid's area is $bh/3$, then all similar pyramids are $bh/3$.

There are full proofs out there, but I hope this gives a better intuitive understanding of why this works.

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Just adding some visual aids for "when you elongate any $3$d shape along one axis by a factor of $u$"(This is not an answer). Suppose you scale the unit pyramid by $u=7$ in $z$ axis. It will look like this. enter image description here

Let's change $z$ axis to $u$ axis where every unit length on vertical axis is $u$ times the unit of $z$. enter image description here

So in this axis our pyramid will have volume $\frac{1*1*1}{3}$ and changing back to original space we get $\frac{u*1*1}{3}$.

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