Everything you ever needed to know about pyramids

In grade 9 we looked at a variety of pyramids and their related prisms (same base and height).  We estimated the number of times the pyramid could be filled with water and emptied into the prism to fill it up.  Guesses ranged from 1.75 times to 3.5 times, with most of the class finally settling around the guess of 2 times.

However minds changed quickly when the pyramid was poured once into the prism.  After that we all agreed that it’d take 3 pyramids to equal the prism volume, or that the pyramid was 1/3 the volume of the prism.

 We set about to build a pyramid with the volume of 300cubic centimeters. We needed to do some calculations, and figure out what dimensions would work.  Many groups figured out dimensions of a prism that has a volume of 900 cubic centimetres to start with.

We then figured out what to cut out of the paper.

Some pyramids were short

Some were tall
And some started to look more like a prism than a pyramid, but the good things about building with old file folders is that you can always start over!

Many groups had to start over, because for many their pyramids were too short.  Many built pyramids hoping for the height to be 9, but instead they measured the height of the sloped triangle sides to be 9, so the entire pyramid height would be smaller than that.

The pythagorean theorem is there, right inside the pyramid.  The height (black) and half of the base (pink) form the legs of the triangle, and the slant height (pink) is the hypotenuse.

h^2+(0.5b)^2=s^2

We had a few good pyramids by the end of class, and we also had a much better understanding of area and volume.