Two Circles and a Square - Problem With Solution

Solution to Problem :

• If x is the size of one side of the square, the small circle has a diameter of size d = x and its radius has a size of x / 2, hence its area A2 is given by
  
  
  
  A2 = Pi (x/2) ² = Pi x ² / 4
  
  
• The radius R of the large circle is equal to D / 2 where D is the diagonal of the square. Pythagora's theorem gives
  
  D ² = x ² + x ²
  
  D = x sqrt (2)
  
  R = x sqrt(2) / 2
  
  
• The area A1 of the large circle A1 is given by.
  
  A1 = Pi (x sqrt(2) / 2) = Pi x ² / 2
  
  
• Hence
  
  A1 / A2 = (Pi x ² / 2) / (Pi x ² / 4) = 2

More references to geometry problems.