Volume of the Pyramid

For some time I asked myself why the volume of the pyramid is 1/3*B*h.

In this post I will give an unusual demonstration of this fact.

After some reflection that the volume is directly proportional to the Area "B" and the height "h" seems reasonable but the 1/3 puzzled me. I know some different demonstrations for it but I wasn't completely satisfied with them.

If you already saw this idea/demonstration somewhere else please tell me where so I can check it.

For some (important) motivation let's first take a rectangular prism with square base ABCD and cut it with a plane not parallel to the base (such that the cut is between the two bases at the "bottom" and the "top"), considering below such a plane a truncated solid that at each vertex ABCD the solid has heights h _a , h _b , h _c , and h _d .

What is the height of the solid at the center of the square H?

You can show using some geometry (and it should appeal to your intuition) that

H = (h _a + h _c )/2 = (h _b + h _d )/2

That is also equal to our idea of the "average height of the solid" and it is also equal to the average at the points A,B,C ,and D.

H =(h _a + h _b + h _c + h _d )/4 .

The volume of the truncated solid V should be

V = (Area of base B)*(average height) "*"

And what happens if the base is an equilateral triangle ABC? The same idea should apply so its volume would be:

V= B*(h _a + h _b + h _c )/3.

Lets analyze the case when h _a = h _b = 0 and h _c = h... in that case we have a pyramid!!! So it's volume should be:

V= B*(0 + 0 + h)/3 = B*h/3

That is it!

Unfortunately this demonstration is not rigorous. I think it is intuitive thought, and it can be made rigorous using integration, or more simply the idea that sheering doesn't change the volume.

Read on for the integration point of view.

There are three equivalent pyramids we can make:

h _a =0, h _b = 0, and h _c = h
h _a =0, h _b = h, and h _c = 0
h _a =h, h _b = 0, and h _c = 0

These pyramids are all equal so they have the same volume, and each pyramid is determined by a different plane. We can think the volume as the integral of the height(function) over the base ABC of the pyramid. If we add the functions determined by each plane we will also add their volumes so we arrive at a linear function where

h _a = h _b = h _c = h

Such is the constant function, so that the three pyramids "added" together equal the volume of the prism and since each has the same volume and so the volume of the pyramid must be 1/3 of the prism with the same base**.

Why go to all that trouble? There is a very well know way to fit three pyramids in a triangular prism. The problem with that is these pyramid are not easy to "see", and we still have to argue why they should have the same volume***.

The main advantage here is the connection of our idea of average and the 1/3 factor of the pyramid. We will also see in "the sums of squares" that this point of view has nice implications.

For those who want more detail see Observation below:

* Obs. One can also demonstrate this by dividing the solid in prisms, or using "reflection maintain the volume" reasoning.

** Obs. I made the base of the pyramid an equilateral triangle to ease things up, we can use Cavalieri's Principle to get to the other cases, and we have too anyway to get the height off a vertex at the base. We can also use to the same effect the principle that pyramids of equal base area and height have the equal volumes.

*** Obs. The three equal pyramids from the corner of a cube or the six equal pyramids from the center, may be easier to see, but still we have to argue why this is true for more general pyramids.