Apothem of a Triangle

Date: 03/21/2001 at 18:50:40
From: Anne Hufnus
Subject: Apothem of triangles

I read your explanation on finding apothems for hexagons; however, 
it does not seem to apply to the triangle. I have a triangle with one 
side measuring a length of 12, and am asked to find the apothem and 
radius.  How would I go about this?

Date: 03/21/2001 at 22:31:28
From: Doctor Peterson
Subject: Re: Apothem of triangles

Hi, Anne.

I presume you are talking about an equilateral triangle, seen as a 
regular trigon. You can actually convert this problem in part to that 
of the  hexagon, by dividing each side in thirds:

            +
           / \
          +---+
         /     \
        +   +   +
       / \  |  / \
      +---+-+-+---+

The apothem is that short vertical line from the center to the 
midpoint of an edge, and is also the apothem of the hexagon. You may 
want to continue what I started, and divide the whole triangle into 
nine small equilateral triangles. If you look closely, you will then 
see that the circumradius of the triangle will be twice the apothem, 
since it consists of the altitudes of two of the little triangles.

If this isn't enough help, just let me know.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/