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### Miter for a Pyramid

Date: 07/31/97 at 10:44:04
From: Clive Champion
Subject: Miter of a pyramid

I am an artist and have a project where I have to construct a 4-sided
pyramid out of glass for a garden fountain. Each of the four triangles
has a base of 30.5 inches and a height of 46 inches. I need to know
what degree of miter to put on the edges of the uprights of the
triangles so that it will all fit together nicely.

Thanks for any help you can give me. I have come up with 54.125
degrees on each of the upright edges. I am sure there must be a
formula but I cannot find it.

Clive@artistcraft.com
http://www.artistcraft.com

Date: 07/31/97 at 13:24:50
From: Doctor Ken
Subject: Re: Miter of a pyramid

Hi Clive -

I'll give your measurements some names: let B be the length of the
base of each triangle, and let H be its height. Here's a diagram:

First, find the length L of the legs (the sides that aren't the base)
of each triangle.  You can either measure it, or use the Pythagorean
theorem:

(B/2)^2 + H^2  =  L^2, so L = Sqrt{(B/2)^2 + H^2}

We'll use this length L in our calculations.

Draw a diagram of your assembled pyramid, and make sure you can see
two adjacent faces of it. Locate the midpoints of two of the bases of
the triangles: call them m and M. Now draw the line segment connecting
them - it's called Mm.

Let's name the point connecting the adjacent bases C, and the top of
the pyramid T.

Now drop a perpendicular from m to TC - it meets TC at J. Note that MJ
is also perpendicular to TC (in a different plane). We want to find
the measure of angle mJM.

To do that, first let's find out everything there is to know about
triangle mJM.  If we find out the length of its sides, we'll be able
to find out everything else.

Actually, I'll let you find out the length of its sides - to find out
mM, notice that it cuts across the square base at a 45 degree angle.
To find mJ (and MJ, since they're the same length), notice that
triangle mJC is similar to triangle TmC, so you can set up some
ratios.

Now that you have these lengths, you can use the law of sines to find
the angle mJM.  The law of sines says that in a triangle ABC (with
side a opposite angle A, b opposite B, and c opposite C),

Sin A     Sin B     Sin C
-----  =  -----  =  -----
a         b         c

Using that, and the fact that the angles add up to 180, you should be
able to find out angle mJM, the angle at which two faces of your
pyramid meet.

If you can't follow this, or get stuck somewhere, write back.  Good
luck!

-Doctor Ken,  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

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