The perimeter of a square and a regular hexagon are equal. The ratio of the area of the hexagon to the area of the square is:
Answer & explanation
Correct answer: option 2
Let the side of the square = a
Perimeter of the square =4a
Let the side of the hexagon =b
Perimeter of the hexagon =6b
Given, 4a=6b
Perimeter of the hexagon =6b
Given, 4a=6b
Thus, a=3b/2
Area of the square =a2
Area of the hexagon =(3√3)b2/2
Now, Area of hexagon : Area of square = (3√3)b2/2 : a2
⇒(3√3)b2/2 : (3b/2)2
⇒ (3√3)b2/2 : 9b2/4
⇒ 3√3/2 : 9/4
⇒ 2 : √3
Area of the square =a2
Area of the hexagon =(3√3)b2/2
Now, Area of hexagon : Area of square = (3√3)b2/2 : a2
⇒(3√3)b2/2 : (3b/2)2
⇒ (3√3)b2/2 : 9b2/4
⇒ 3√3/2 : 9/4
⇒ 2 : √3
The correct answer is Option (2) → $2:\sqrt{3}$