A-frame for Swing

Date: 07/26/97 at 12:07:31
From: Dennis Ribblett
Subject: A-frame for swing out of 4 by 4's

I am trying to build a stand for a swing.  The ends are A-frames out 
of 4 by 4's. The brace for the top is a 4 by 4. The legs are to be 
4 feet apart and the length of each 8 feet. 

I need to miter the bottom of each of the 8 foot sections and the top 
of each to join to the 4 by 4 brace. What is the angle of the miter 
cut?

Date: 07/29/97 at 10:06:15
From: Doctor Rob
Subject: Re: A-frame for swing out of 4 by 4's

Let's see.  Let this be an approximate picture:

                __ Q
               |  |
              /|__|\
             /  /\  \
            /  /  \  \
           /  /    \  \
          /  /      \  \
         /  /        \  \
        /  /          \  \ 8 feet
       /  /            \  \
      /  /              \  \
     /  /                \  \
    /  /                  \  \
   /  /                    \  \
  /  /                      \  \
 /  /                        \  \
----                        R ---- P
    <--------- 4 feet ------->

I put the angle at point P equal to A.  Then from looking at similar
triangles (the big one a right triangle with hypotenuse PQ = 96, the
small one a right triangle with horizontal hypotenuse PR), I get the
equation

     4
   ----- + 22
   sin A
   ---------- = cos A.
       96

A numerical solution produces the answer A = 74.18859 degrees, or, for
your purposes, 74 degrees. It turns out that PR = 4.15730 inches, or
nearly 4+5/32 inches.

Of course, this is assuming that your 4-by-4s are really 4.00000 
inches square, which is very likely to be false. Nevertheless, this 
answer will be very close to correct.

By the way, I think I would put the feet farther apart for safety.  
If they are 8 feet apart, and therefore the end triangles are 
equilateral, the mitre angle would be 60 degrees. Of course this 
depends on how firmly the feet are anchored.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/