Inscribed Circle

Date: 18 Mar 1995 14:09:31 -0500
From: Anonymous
Subject: formula for the area of an inscribed circle in 
              an equilateral triangle

I need the formula to find the radius of an inscribed 
circle in an equillateral triangle. I also need to know 
the rationalization for the method.

Date: Sun, 19 Mar 1995 10:19:16 +0000
From: Dr. Math
Subject: Re: formula for the area of an inscribed circle 
              in an equilateral triangle

Hello there!

Here's one way to figure out how long the radius in 
question is:  let's say we have an equilateral triangle, 
and each side has length s.  Then find the center of the 
triangle.  You can do this by drawing the three altitudes, 
or the three medians; in this case, they're all the same
lines.  Then draw the segment that starts at the center 
of the triangle and ends at the midpoint of one of the 
sides.  This is the radius of the inscribed circle.

To find out how long it is, notice the following:  Let's 
say our triangle has vertices X, Y, Z, and let's call the 
center of the triangle C, and the midpoint of one of the 
sides M.  Then look at triangle XCM.  Since each angle 
in an equilateral triangle is 60 degrees, how big is angle 
MXC?  So, is this a special kind of triangle (Hint: yes!!)?  
So, if it has its longer leg with length s/2, you can use 
that information to find the length of its shorter leg.

Let us know if you're still stuck!

-Ken "Dr." Math