SOLUTION: A circle is inscribed in a square. What is the probability to the nearest thousandth that a point inside the square is also inside the circle?

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Question 387154: A circle is inscribed in a square. What is the probability to the nearest thousandth that a point inside the square is also inside the circle?
Answer by Edwin McCravy(20055) About Me  (Show Source):
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A circle is inscribed in a square. What is the probability to the nearest thousandth that a point inside the square is also inside the circle? 


The area of the circle is Acircle = pr2, where r is the radius of the circle.

The area of the square is Asquare = s2, where s is a side of the square
                      Acircle  
So the probability = --------- 
                      Asquare    

That is, 
                   pr2 
The probability = -------
                    s2  


Draw a radius of the circle:



A side of the square is twice as long as the green radius, so s = 2r 

Substitute (2r) for s in:

                    pr2 
The probability = -------
                    s2 

                    pr2 
The probability = -------
                   (2r)2 

                    pr2 
The probability = -------
                   22r2

The r2's cancel:

                    p 
The probability = -----
                   22
 
                    p 
The probability = -----
                    4

                   3.1416 
The probability = -------- = 0.785
                     4


Edwin