Let ABCDEF be a regular hexagon with each side of length 1 cm. The are

Question

Let ABCDEF be a regular hexagon with each side of length 1 cm. The area (in sq cm) of a square with AC as one side is

A. √3
B. 3
C. 4
D. 3√2
E. 8

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CareerGeek · Answer

Formula: Each interior angle in an n-sided regular polygon =  \frac{180(n - 2)}{n}

For regular hexagon, Each interior angle =  \frac{180(6 - 2)}{6} = 120

Triangle ABC is isosceles with angles (30, 30 ,120)
--> Ratio of sides AB : BC : AC =  1 : 1 : \sqrt{3}

Given, AB = BC =  1 . So, AC =  \sqrt{3}

--> Area of square with side AC =  (\sqrt{3})^2 = 3

Option B

GMATWhizTeam · Answer

Solution: 
 o ABCDEEF is a regular hexagon
 o Every sides of the hexagon are equal.
o Every angles of the hexagon are equal.  
Now, join A and C and draw a perpendicular of BM on AC
 https://www.dropbox.com/s/q82ipjlncvbpzip/40.jpg?dl=1 
 o We know that ∠ABC = 120  
 o ∠ABM = 60   
o Now, triangle ABM is a 30: 60 : 90 triangle
 o  BM: AM: AB = 1: \sqrt{3}: 2  
    \frac{AM}{AB} = \frac{\sqrt{3}}{2} 
  \frac{AM}{1} = \frac{\sqrt{3}}{2} 
  AM =\frac{\sqrt{3}}{2}   
o  AC = AM + CM = \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} = 2*\frac{\sqrt{3}}{2} = \sqrt{3} 
o Area of the square =  (\sqrt{3})^2 = 3  
Hence, the correct answer is  Option B.

Let ABCDEF be a regular hexagon with each side of length 1 cm. The are

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Let ABCDEF be a regular hexagon with each side of length 1 cm. The are

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