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
The correct answer is Option (2) → \(2:\sqrt{3}\)
With the same perimeter, the sides are found from it first. A regular hexagon is made of six equilateral triangles, so its area is \(\displaystyle 6\times\frac{\sqrt{3}}{4}a^2\) for side a.
Assuming the common perimeter to be 12:
Square: side \(=12\div4=3\), area \(=9\)
Hexagon: side \(=12\div6=2\), area \(\displaystyle =6\times\frac{\sqrt{3}}{4}\times2^2=6\sqrt{3}\)
The ratio:
Hexagon : square \(=6\sqrt{3}:9=2\sqrt{3}:3\)
Since \(3=\sqrt{3}\times\sqrt{3}\), the ratio \(2\sqrt{3}:3\) is the same as \(2:\sqrt{3}\)