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?

{{{drawing(200,200,-10,10,-10,10,

rectangle(-9,-9,9,9), circle(0,0,9) )}}}
<pre><font size = 4 color = "indigo"><b>
The area of the circle is A<sub>circle</sub> = <font face = "symbol">p</font>r<sup>2</sup>, where r is the radius of the circle.

The area of the square is A<sub>square</sub> = s<sup>2</sup>, where s is a side of the square
                      A<sub>circle</sub>  
So the probability = --------- 
                      A<sub>square</sub>    

That is, 
                   <font face = "symbol">p</font>r<sup>2</sup> 
The probability = -------
                    s<sup>2</sup>  


Draw a radius of the circle:

{{{drawing(200,200,-10,10,-10,10,
green(line(0,0,9,0)),
rectangle(-9,-9,9,9), circle(0,0,9) )}}}

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

Substitute (2r) for s in:

                    <font face = "symbol">p</font>r<sup>2</sup> 
The probability = -------
                    s<sup>2</sup> 

                    <font face = "symbol">p</font>r<sup>2</sup> 
The probability = -------
                   (2r)<sup>2</sup> 

                    <font face = "symbol">p</font>r<sup>2</sup> 
The probability = -------
                   2<sup>2</sup>r<sup>2</sup>

The r<sup>2</sup>'s cancel:

                    <font face = "symbol">p</font> 
The probability = -----
                   2<sup>2</sup>
 
                    <font face = "symbol">p</font> 
The probability = -----
                    4

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


Edwin</pre>