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/