Triangle and Circle with same Center

Date: 7/8/96 at 9:6:16
From: Anonymous
Subject: Find the length of the side of the triangle

Dear Dr. Math,

An equilateral triangle and a circle have the same center.  The 
area of that part of the triangle not inside the circle equals the 
area of that part of the circle not inside the triangle.  If the 
radius of the circle is 1, find the length of a side of the triangle 
(to the nearest tenth of a unit.)

Thanks for your help.

Nancy Geldermann

Date: 7/8/96 at 22:41:0
From: Doctor Pete
Subject: Re: Find the length of the side of the triangle

Call the area inside the triangle not inside the circle area A, and 
the area inside the circle not in the triangle area B.  In addition, 
call the area inside both the triangle *and* the circle area C.  
Since A+C is the area of the triangle, and B+C is the area of the 
circle, it follows that if A = B, then A+C = B+C, or the area of the 
triangle must be equal to the area of the circle.  It follows that 
since the area of the circle is Pi(1)^2 = Pi, the area of the 
triangle is also Pi.  Since the triangle is equilateral, its area in 
terms of a single side is (Sqrt[3]x^2)/2 where x is a side.  Thus 
Sqrt[3]x^2 = 2 Pi, or

              [  2 Pi   ]
     x = Sqrt [ ------- ] , or about 1.9046256.
              [ Sqrt[3] ]

-Doctor Pete,  The Math Forum
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