Geometry in Architecture - what is the height which maximize

Geometry in Architecture - what is the height which maximize

Postby Math Tutor » Tue Apr 29, 2014 8:00 am

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?

image1.png
image1.png (7.25 KiB) Viewed 1642 times

For a right rectangular pyramid, the enclosed volume is:

image2.png
image2.png (2.14 KiB) Viewed 1642 times


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

image3.png
image3.png (4.78 KiB) Viewed 1642 times


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?


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Re: Geometry in Architecture - what is the height which maxi

Postby Guest » Thu May 01, 2014 4:43 am

This is a badly formed question.

The volumetric efficiency of a sphere is r/3 so to maximize it we should take r as large as possible.
Similarly we can argue that taking the height as large as possible maximizes the volumetric efficiency of a pyramid (square base or not and including the base or not).

This is probably not what you are after. You probably want it to be scale invariant. However, you define the scaled volumetric efficiency of a sphere to be 1/3 by setting r=1. If instead we had chosen to characterize spheres via diameters instead of radii then the VE would be d/6 and setting d=1 gives a different scaled VE of 1/6. So the scaled VE depends on the parameter you choose to set to 1. For a pyramid you'll get different answers depending on whether you set w=1 or, l=1, or l+w=1, etc.

Hope this helped,

R. Baber.
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Re: Geometry in Architecture - what is the height which maxi

Postby Guest » Mon May 05, 2014 5:26 pm

For the special case H = L = W = R for pyramid V = (R^3) / 3
Perp Height of Sloping side is (5R^2) / 4
Surface area one side = [(5R^3) / 8]
So for 4 sides = [(5R^3) / 2]
Plus bottom = [(5R^3) / 2] + R^2
Efficiency as V / A = [(R^3) / 3] / [[(5R^3) / 2] + R^2]
= 1/ [7.5 + (3/R)]

As R goes from 0 to infinity ...Effic. goes from infinity to 1/7.5
For case R=1 ..... then Effic. = (1 /10.5)
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