In the book "Geometry in Architecture", William Blackwell discusses the efficiency of various solids in terms of the amount of enclosed volume for a given surface area.

He notes that a sphere is the shape for which the most volume is enclosed by a given surface area and considers other solids to see how they compare to a sphere in this regard.

Define the volumetric efficiency as the ratio of the enclosed volume to the surface area.

For a sphere, the volume enclosed is 4/3*pi*r^3 and the surface area is 4*pi*r^2.

And the volumetric efficiency of a sphere (VEs) is (4/3*pi*r^3)/(4*pi^r^2) = r/3

Scaling the solution by setting the radius equal to 1 unit of length, then the VEs of all spheres is the same, 1/3.

What about a right rectangular pyramid as pictured below?

For a right rectangular pyramid, the enclosed volume is:

while the area of the pyramid (including the base rectangle) is:

What is the height which maximizes VEp for the special case of a square base (l = w)?

What is the height which maximizes VEp for the general case of any rectangular base?

What is the height that gives the maximum VEp for the general case of any rectangular base but where only the area of the four triangular sides of the pyramid are considered without including the area of the base?

Sent by John Marcou

via email.