Try this beautiful problem from Geometry: Area of the 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?
Geometry
Triangle
Hexagon
Answer: \(6\)
AMC-8 (2012) Problem 23
Pre College Mathematics
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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