Associated Topics || Dr. Math Home || Search Dr. Math

### Pyramid Construction

```
Date: 05/24/97 at 02:39:31
From: Don Mohr
Subject: Pyramid building

I'm not a student - I am a professor of history. My son is an advanced
math tenth grader. Neither he nor his teacher could help me.

I need to construct a four-sided pyramid for a children's exhibit at
the Anchorage Museum. I want to use plywood. I need to figure out the
compound angles (angle of side, and mitre) that will join the sides.

After a much-needed review of trigonometry (and with my son's help),
I was able to figure out the mitre angle needed.  The angle of the
sides of the pyramid are directly calculable by the Pythagorean
theorem. I found the mitre angle (joints between sides) by laying out
a right angle triangle on a side one foot from the point of the base.
Knowing the side angle, one foot length of side, and ninety degree
angle allows the calculation of the other sides. Then looking down at
a corner of the base from above the pyramid, one has a triangle that
is perpendicular to the joining edge of two sides. The length of all
sides of this triangle are known. The angle of the joined sides may
be calculated and the mitre revealed.

My son has even programmed his TI-85 to calculate the compound mitre
of any four-sided pyramid by entering only the base length and height.

Now my question is this:

It seems to me that there should be a more direct path to calculating
the angle of the sides. It appears that the angle of the sides must
change in a linear fashion determined by the base length and height.
Is there a formula that will do this?  Once this angle is known, the
mitre calculation can be done in one's head. Thus, one would not have
to carry around a TI-85.

```

```
Date: 05/24/97 at 07:57:10
From: Doctor Mitteldorf
Subject: Re: Pyramid building

Dear Don,

A picture would really help here! I gather that the pyramid you are
working with has 4 triangular faces and a square base (as opposed to a
tetrahedron, which has 4 triangular faces, including the base).

A square pyramid like this can be constructed with four copies of any
isoceles triangle, as long as the height of the triangle is more than
half its base. Once you decide the dimensions of the isoceles
triangle, you can use the fact that the apex is above the center of
the square base to calculate the mitre angle of the base: the cosine
of this angle is half the base of the triangle divided by the height
of the isoceles triangle.

The only other angle left to compute is the mitre angle where any two
adjacent triangular faces meet. There's a formula I carry around in
my head that is useful in this and many situations like it, involving
angles between lines and between planes. I'll describe it to you, with
three special cases:

Open a notebook and draw a line from a point on the book's spine
diagonally across one page. Draw another line starting at the same
point on the spine, going diagonally across the facing page at a
different angle. Let's say that the angle between the spine and the
first line is a and the angle between the spine and the second line
(measured on the facing page) is b. Let x be the angle between the two
lines.

Now this angle x will depend on how wide open the book is.  If the
book is open flat, then the angle x becomes a + b. If the book is
closed tight, the angle x becomes a - b. The interesting case, of
course, is when the book is open to an intermediate angle. If x is a
right angle so that the book is open to 90 degrees, then
cos(x) = cos(a)*cos(b)

You can write the other two cases in terms of cosines as well:

cos(x) = cos(a)*cos(b) - sin(a)*sin(b)     Book open flat
cos(x) = cos(a)*cos(b) + sin(a)*sin(b)     Book closed tight

These come from the formulas for the cosine of the sum and difference
of two angles.

Now here comes the general case: For any angle c that the book may be
open:

cos(x) = cos(a)*cos(b) + sin(a)*sin(b)*cos(c)

Notice that this formula subsumes the three special cases we did
first. The general formula can be derived easily using vector dot
products. It can also be derived straight from trigonometry, if you
have a lot of imagination and are good at visualizing 3-dimensional
objects.

I leave it to you to calculate your pyramid angles using this formula.

-Doctor Mitteldorf,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry
High School Practical Geometry

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search