What is volume formula for any-base oblique cone and pyramid?

caasdads 06/12/2018 at 14:54. 3 answers, 452 views
geometry volume

Does the formula $V = 1/3 \cdot S\cdot h$ also apply to:

  1. Skewed (oblique) cone with any-shaped base - including elliptic base and any other figure imaginable base (which area is $S$) and height (vertically) is $h$
  2. Skewed (oblique) pyramid with any polygon-shaped base (including irregular polygon base)?

3 Answers

Lucas Resende 06/12/2018 at 15:22.

I'll assume you are ok with basic calculus, let's use integration to show that it's true. Put the base of the solid on the $xy$ plane at $z=0$ and the top at $z = h$, if we cut at $z=x$, $x\in[0,h]$, the area of the slice will be $S\left(\frac{h-x}{h}\right)^2$ cause the area decline with the square of the height, to get the volume we just integrate: $$ V = \int_{0}^h S\left(\frac{h-x}{h}\right)^2 dx = \int_{0}^h \frac{S}{h^2}\left( h^2 - 2hx + x^2 \right) dx = \left.\frac{S}{h^2}\left(h^2x - hx^2 +\frac{x^3}{3}\right)\right|_0^h = \frac{S}{h^2}\left(h^3-h^2+\frac{h^3}{3}\right) = \frac{Sh}{3}$$

As we want the volume is $1/3 \times S \times h$, where $S$ is the area of the basis (being the basis whatever you want) and $h$ the height.

gimusi 06/12/2018 at 15:00.

Yes it is true in general for generic right cones or pyramids and it is also true for oblique cones and pyramids by Cavalieri's principle.

Acccumulation 06/12/2018 at 21:31.

If each cross section is similar to every other cross section, then each cross section can be described as being congruent to the base after some scaling factor $s$ has been applied. The area of a cross section will then be $s^2$ times the area of the base S.

Clearly, at the top, $s = 0$. At the bottom, $s = 1$. The volume of a figure can be considered to be the sum of an infinite number of slices of the figure. The volume of a slice is equal to the cross sectional area times the height element.

$V = \int_0^h A(h)dy = \int_0^h s^2Sdy $

The scaling factor goes from 1 to 0, a total change of -1. $y$ goes from 0 to h, a change of h. If the scaling factor is scaling linearly, then $dy = -hds$.

$V = \int_1^0 -hs^2Sds = \int_0^1 hs^2Sds = hS\int_0^1 s^2ds = \frac{hS}{3} $

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