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Is the proof on the volume of a generalzing the volume of a square pyramid correct

Introduce a $z$ axis perpendicular to the base, and positive in the downward direction, with the origin at the tip of the pyramid.

Let the area of the base of the pyramid be $B$. For $0<z<h$, a cross section parallel to the base of the pyramid is similar to the base. Since the side lengths are $\propto z$, the area are $\propto z^2$.

$$\text{Area of Cross Section at a depth of z = } A(z) = B\times\frac{z^2}{h^2}$$

Now, using the standard Calculus formula for volume of a solid: $$\text{Volume} =\int_{0}^{h}{A(z)}dz =\int_{0}^{h}{\left(B\times\frac{z^2}{h^2}\right)}dz =\frac{B}{h^2}\int_{0}^{h}{z^2}dz =\frac{B}{h^2}\times \frac{h^3}{3} =\frac{Bh}{3} $$

which is the standard formula for volume of a pyramid

Starlight
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  • Compare this answer to another question. They wrote $H$ where you wrote $h$ and they wrote $h$ where you wrote $z$. The one thing that answer has that is not so clear from your proof is where the denominator $h^2$ comes from. – David K Dec 20 '21 at 14:03
  • @DavidK That is very useful. I edited it to attempt to improve the clarity. Does the rest of it look ok? – Starlight Dec 20 '21 at 14:06
  • You hold $h$ constant while the parallel cross sections vary in size. (I'm almost sorry that I linked to a question that uses $h$ differently.) I think you mean to say the side lengths are proportional to $z$ and the areas proportional to $z^2.$ That means the area is $kz^2$ for some constant of proportionality $k.$ In particular, $k = B/h^2$ ... why? Because you need something that gives the result $B$ for the particular cross section at $z=h$, which happens also to be the base of the entire pyramid. – David K Dec 21 '21 at 03:01
  • @DavidK You are correct. I did not pay attention when writing (and that is a cardinal error). It is indeed proportional to z and z^2, respectively. – Starlight Dec 21 '21 at 03:57
  • Does this answer your question? Volume of Pyramid – Kurt G. Jan 22 '23 at 07:37
  • My question was to verify my proof.... – Starlight Jan 23 '23 at 00:20

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