MathematicsThe side of a regular hexagon is 2 cm. Then, what is the ratio of the radius of the circumscribed circle to the radius of the inscribed circle?

The side of a regular hexagon is 2 cm. Then, what is the ratio of the radius of the circumscribed circle to the radius of the inscribed circle?


  1. A
    2 3  
  2. B
    3  
  3. C
    2  
  4. D
    1 3   

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    Solution:

    Let the diagram of the given situation be,
    imageFrom the above figure, EO is the diameter of the circumscribed circle.
    In triangle EMF, we have
    sin 30 = EM EF (EFM= 30 ) 1 2 = EM 2 EM=1  
    O is the center of the hexagon. Then,
    OM= AF 2 (AF=2) OM=1  
    Then, the radius of the circumscribed circle is,
    OE=EM+OM =1+1 =2  
    From the figure, we can say that PO is the radius of the inscribed circle. Since PO=FM  , then
      cos 30 = FM EF 3 2 = FM 2 FM= 3  
    The ratio of the radius of the circumcircle and incircle is given by OE PO = 2 3  .
    Hence, the correct option is 1.
     
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