Building a Wooden Square-Based Pyramid

Date: 01/26/2001 at 20:23:21
From: Meg Hosler
Subject: Building a wooden Square-based Pyramid 

I want to build a wooden pyramid that has a bottom square floor of 
8"x8". If you could just tell me the measurements and angles in inches 
of one side and the bottom, then I could just make three more and have 
it right. The bottom is 8"x8" and the other triangles should have 
all three sides the same I think, but what are the angles and lengths?  

If you could use your example from the FAQ and actually fill in the 
numbers where you have symbolic words, I could perhaps deduce what the 
diagram is showing.  

I look forward to your response. Thanks.
Meg

Date: 01/27/2001 at 07:24:22
From: Doctor Mitteldorf
Subject: Re: Building a wooden Square-based Pyramid 

Dear Meg,

My favorite formula for doing this kind of problem is one I call the 
"hinge formula." Imagine drawing two angles, a and b, one on each face 
of a hinge. The angles share the hinge axis as a common ray. What is 
the angle between the outer rays of angles a and b?  The answer is

   cos(c) = cos(a)cos(b) + sin(a)sin(b)cos(phi)

where phi is the angle that the hinge is open to.  

In your case, think of the hinge as connecting the triangular side to 
the square bottom. Since the triangular side has three equal legs, its 
angles are all 60 degrees. Since the bottom is a square, its angles 
are all 90 degrees. So at the bottom corners, we have a = c = 60 and 
b = 90. 

This allows us to calculate phi. Phi is the mitre angle that you need 
to attach the sides of the pyramid to the bottom. By the above 
formula,

   cos(phi) = [cos(c) - cos(a)cos(b)] / [sin(a)sin(b)] = 1/sqrt(3)

   phi = 54.7 degrees.

The other mitre angle you'll need is the angle between adjacent 
triangular walls. This comes from the same hinge formula, but viewed 
from a different direction, so that now the hinge is on the side. 
So a = b = 60 and c = 90.

   cos(phi) = [cos(c) - cos(a)cos(b)] / [sin(a)sin(b)] = -1/3

   phi = 109.5 degrees.

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/   

Date: 01/28/2001 at 02:08:57
From: Meg Hosler
Subject: Re: Building a wooden Square-based Pyramid

Thank you. I have studied this and I think if I divide the numbers 
you gave me by 2 it will be the proper setting for cutting the angles 
on the wood base and triangles. In other words, I will make three 
triangles of equal sides 8" in length, with a miter cut at 54.7 
degrees for the total angle when joined together with the adjacent 
base side, and 27.3 degrees for each angle on the wood triangle 
adjoining side to the next triangle. 

I hope this works - if not, the plywood mockup will teach me 
something, and then I can try again.

Thank you again.

Date: 01/28/2001 at 07:15:24
From: Doctor Mitteldorf
Subject: Re: Building a wooden Square-based Pyramid

Meg -

That's right - if you have the angle between two planes, then you can 
divide that angle in half and cut both pieces with the mitre saw set 
to half that angle. (For example, cutting two boards at 45 degrees 
will create pieces that fit together at a right angle.)

This may seem like a lot of theory - will you let me know if it works?

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/