# Create a 3D object to demonstrate the pyramid volume equation [closed]

Under the assumption that every triangle area is given by the equation "S = constant X Base X height", with a simple drawing it is demonstrated that the constant is 1/2. Assuming that a pyramid volume is given by "V = constant X base X height" create an object that shows that the constant is 1/3 for the pyramid volume. This is not requiring math knowledge - it requires the ability to imagine a 3D object that demonstrates (no proof - this is why this a puzzle for enjoyment) the feature of pyramids aka "equation".

I will describe a method

Take a cube and connect each vertex to the centre via a straight edge. We readily see that the picture is as if we have taken six square pyramids whose bases match the faces of the square and whose height is half the edge of the cube. This readily gives us the constant for a pyramid with a square base as being 1/3.

To prove that this applies to a pyramid with triangular base notice that we can slice the square based pyramid with a plane perpendicular to the square base which slices it along the diagonal. This divides the square pyramid into two congruent triangular pyramids and thus each has half the volume, half the base and the same height so the constant is still 1/3.

• Could you get a direct slicing of a cube into 3 pyramids? Could you draw it?
– Moti
Commented Jun 7, 2023 at 14:57
• @Moti actually this method would be represented by a slicing of the cube into 12 pyramids whose bases are half the face of the cube and whose heights are half the edge length. Commented Jun 8, 2023 at 8:45
• Could you dissect the Cube into 3 identical pyramids?
– Moti
Commented Jun 9, 2023 at 14:58
• @Moti yes you are right, there is such a construction, see here for example: math.brown.edu/tbanchof/Beyond3d/chapter2/section02.html Commented Jun 9, 2023 at 16:02
• This is the elegant and clear solution! BTW it applies to higher dimensions.
– Moti
Commented Jun 9, 2023 at 20:00