A square of area 64 cm2 is inscribed into a semi-circle. What is the area of the semi-circle?


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Answer:

40π cm2

Step by Step Explanation:
  1. The following figure shows the square inscribed into a semi-circle,

    Let's assume, a is the length of the side of the square.
    Therefore, AB = BC = CD = DA = a,
    The area of the square = a2
  2. According to the question, the area of the square is 64 cm2.
    Therefore, a2 = 64 -----(1)
  3. If we look at the figure carefully, we notice the OC is the radius of the semi-circle and 'O' is the center of the semi-circle.
    Therefore, OA = OB =  
    a
    2
     
  4. In right angled triangle OBC,
    OC2 = OB2 + BC2  [By the Pythagorean theorem.]
    = ( 
    a
    2
     )2 + a2
    =  
    a2
    4
      + a2
    =  
    5a2
    4
     
    =  
    5 × 64
    4
       [From equation (1)]
    =  
    320
    4
     
    = 80 cm2
  5. Now, the area of the semi-circle =  
    π(OC)2
    2
     
    =  
    π × 80
    2
     
    = 40π
  6. Hence, the area of the semi-circle is 40π cm2.

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