The frustum can be seen as a Rorschach test. Some people see a gold bar, and some people see an unfinished pyramid.
One way to create a frustum in TINKERCAD is to start with a solid pyramid basic shape. The top part of the pyramid is then removed by a “hole” that adds negative volume to the shape. The result is a frustum. The following video demonstrates this method:
Before removing the top part of the pyramid, we need to know the dimensions of the solid pyramid’s base. We also need to know the height of the pyramid.
We can do this by using our prior knowledge and best practices. One of the most basic uses for algebra is providing a simple way to express how two things add up to one thing.
We know that a frustum is the “bottom section” (in blue) of a pyramid. This means that if we add the missing “top section” (in pink) to the frustum, we will have a complete pyramid.
That relationship can be described by the following equation:
We know the formula for the volume of the complete (big) pyramid,
with the base area B, the height of the frustum h, and the unknown height of the top pyramid H.
We also know the volume of the top section of the complete pyramid (which can also be described as a mini pyramid),
with the base area A (the same area as the top area of the frustum), and the unknown height of the top pyramid H.
The volume of the bottom section of the complete pyramid can be described as a frustum.
With a base area B of 6″ x 6″. let us suppose that the height of the frustum h is 10 inches, and the top area of the frustum A is 2″ x 2″. This means that the total height of the complete pyramid H + h provides us clues to the information that we need finish the pyramid.
To help finish the pyramid, we now have a useful equation that can be re-used and re-purposed in different workplace contexts:
In Wolfram Alpha, type this:
h=10;A=2*2;B=6*6;(1/3)(H+h)*B=(1/3)A*H + (h/3)(A+B+SQRT(A*B))
Notice that we never actually calculated the volumes in the equations. Instead, we used the additive relationships between the volumes, areas and heights to complete a specific workplace task.
The 3D models related to the “finish the pyramid” math practice activity are available on GitHub. Also, the 3D models can be directly imported into TINKERCAD. NOTE: Model units are in inches.