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### 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/
```
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry
High School Practical Geometry

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