Expert: Paul Klarreich - 4/24/2007

Question
Hi, i'm having real trouble with a problem, A circle of radius (r) is inscribed in an equilateral triange and asks to compute the area of the shaded region or otherwise outside of the circle which is in the triangle.  There are no figures given.  

Answer
Questioner:   Matt
Category:  Advanced Math
 
Question:  Hi, i'm having real trouble with a problem, A circle of radius (r) is inscribed in an equilateral triangle and asks

>> I didn't know circles could talk.

to compute the area of the shaded region or otherwise outside of the circle which is in the triangle.  There are no figures given.
.........................................................
Hi, Matt,

If there are no figures given, you make your own.  Draw the Eq. Tri.  Then construct the altitude from each vertex to the opposite side.  That's three of them and they meet at a point, call it C.  Now the distance from C to each of the sides is your radius.  Your diagram will look like this:

USE COURIER FONT TO VIEW THIS.
       B
      /|\
     / | \
    /  |  \
   /   |   \
  /   C+    \
 /     |     \
/      |      \
--------------- A
      D

Also draw in CA on your paper.  Now you will find that:

Angle CAD = 30 degrees.
Angle CDA = 90
CD = r
AC = 2r (and so does BC)
AD = r sqrt(3)

Now you can compute the area of the triangle.  

Each side of the triangle is 2AD - 2r sqrt(3)
The altitude is BD = BC + CD = 2r + r = 3r

The area is 1/2 bh = 1/2(3r)(2r sqrt(3)) = 3r^2 sqrt(3)

The circle is r^2, so the area you want is the difference.

3r^2 sqrt(3) - r^2