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### 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
```
Associated Topics:
High School Trigonometry

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