Question 139193
The formula for the volume of a prism is {{{V=Bh}}} where {{{B}}} is the area of the base and {{{h}}} is the height.


The formula for the volume of a pyramid is {{{V=(Bh)/3}}} where {{{B}}} is the area of the base and {{{h}}} is the height.


So a pyramid with a base equal in area to a prism and a height equal to the height of the prism has {{{1/3}}} the volume.  Notice that the bases don't necessisarily have to have the same shape -- just the same area.  For example, if you had a prism with a triangular base where the base of the triangle was 2 and the altitude of the triangle was 1 and further the height of the prism was 5, then a square based pyramid with sides on the base of 1 unit and a height of 5 would have exactly {{{1/3}}} the volume of the prism.


By the way, this same relationship holds between a cylinder and a cone.  Equal base areas and equal heights implies a 3 to 1 ratio in the volumes.  That makes perfectly good sense if you consider that a cone is just a pyramid with an infinite number of infinitessimally thin faces, and a cylinder is a prism with an infinite number of infinitessimally thin faces.


My 5th grade teacher (more years ago than I care to reveal) demonstrated this relationship.  She had a set of clear plastic geometric solids.  She filled the pyramid with sand three times and it filled the equal-based, equal-height prism exactly.