Question

Four circles are inscribed in squares. Write a function f in terms of the radius r that represents the area of the shaded region.

Answer 1

Answer:

The answer is = 3.432r²

Step-by-step explanation:

The pictorial representation of the question is given below

                       (The first uploaded image)

Step 1: State the parameters and the required formula

  Let L denote the length of the square  

 Let B denote the breath of the Square  

 $Let D denote the diameter of the circle = 2r$

$Let r denote the radius of the circle = D÷2$

    $Area of a square = L × B = L × L = L²$    (Note for a square L = B)

    $Area of circle = πr²$  

Step 2: Consider the first inscription of circle inside the square

    From the diagram we see that the diameter of the circle is equal to the length and breadth of the square.

   Hence the area of the square in terms of radius is  

          $Area of square = D * D$

                                     $= 2r * 2r$

                                    $= 4r²$  

Now to obtain the area of the shaded region for the first inscription of circle we subtract the area of the square from the area of the circle  

          $= 4r² - πr²$          Note: π =3.142

              Therefore we have Area of shaded region for first inscription of circle, $S = 0.858r²$  

Step 3: Now we consider the area of the shaded region for the inscription of  four circles into squares, as shown on the diagram (uploaded image).  

        Since the area of the shaded region for the first inscription is 0.858r² then for the four inscription of the circles into squares,  

The area will be 4 multiplied by area of the shaded region for the first circle inscribed to a square  

i.e. $4 × 0.858r² = 3.432r²$  

Therefore the area of the shaded region is given as 3.432r²