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

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

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} $

- Volume of Pyramid
- Why isn't the volume formula for a cone $\pi r^2h$?
- How to find a missing radius in the surface area formula for a cone with just surface area number, and slant height?
- Where am I wrong at deriving the formula of volume of cone?
- How would the volume of a frustum with irregular polygon area be calculated?
- A right elliptical cone is 4m high and has an elliptical base with half axes lengths of 1m and 2m. Find its volume.
- Find volume of pyramid given points and base
- Volume of a cone inside an upside-down pyramid
- How to show that the volume of a cone with an arbitrary base is $(1/3)Bh$?
- How to find the scale factor of the cross section in a general cone

- Draw a demonstration of Pythagorean theorem with TikZ
- Simple harmonic oscillator by operators
- "Integral Milking:" Does anyone else do this?
- How to choose a good grading curve for yes/no exams?
- What’s a possible one-word replacement for “applicable in every situation”?
- How a led in a ledstrip like ws2811 know when it should be on or off
- Simple question about probability
- New to bicycle riding. Questions about adjustments and upgrades
- Integers sorted by their digital roots
- Players get angry if anything negative happens to them
- What does it mean to "Burn a Zero Day"?
- In a spinning Dyson ring, would objects not touching the surface experience gravity?
- How to plot binary (presence/absence - 1/0) data against continuous variables
- What could people notice about someone who is two times as dense as a regular person?
- How long to save losing lottery tickets (USA)
- Why doesn't this example of basis change work?
- How to deal with unknown genders in English?
- Is it appropriate to ask a software developer about extra hours spent for side projects and open-source (as a hobby)?
- Can I safely eat whatever goes through the x-ray machine?
- Being alive today: the most improbable coïncidence?
- How does electrical power relate to Ohm's law?
- How can I ask my girlfriend to split gas money for a long trip?
- Is this a useless group?
- Determine whether to push or update object in array based on unique ID