Area of a Regular Hexagon | AMC-8, 2012 | Problem 23

An equilateral triangle and a regular hexagon have equal perimeters. If the triangle's area is 4, what is the area of the hexagon?

To find out the area of the Regular hexagon,we have to find out the side length of it.Now the perimeter of the triangle and Regular Hexagon are same....from this condition you can easily find out the side length of the regular Hexagon

Can you now finish the problem ..........

Let the side length of an equilateral triangle is\(x\).so the perimeter will be \(3x\) .Now according to the problem the perimeter of the equiliteral triangle and regular hexagon are same,i.e the perimeter of regular hexagon=\(3x\)

So the side length of be \(\frac{3x}{6}=\frac{x}{2}\)

can you finish the problem........

Now area of the triangle \(\frac{\sqrt 3}{4}x^2=4\)

Now the area of the Regular Hexagon=\(\frac{3\sqrt3}{2} (\frac{x}{2})^2=\frac{3}{2} \times \frac{\sqrt{3}}{4}x^2=\frac{3}{2} \times 4\)=6

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