2012 AMC 8 Problems/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?
Let the perimeter of the equilateral triangle be . The side length of the equilateral triangle would then be and the sidelength of the hexagon would be .
A hexagon contains six equilateral triangles. One of these triangles would be similar to the large equilateral triangle in the ratio , since the sidelength of the small equilateral triangle is half the sidelength of the large one. Thus, the area of one of the small equilateral triangles is . The area of the hexagon is then .
Let the side length of the equilateral triangle be and the side length of the hexagon be . Since the perimeters are equal, we must have which reduces to . Substitute this value in to the area of an equilateral triangle to yield .
Setting this equal to gives us .
Substitute into the area of a regular hexagon to yield .
Therefore, our answer is .
Let the side length of the triangle be and the side length of the hexagon be . As explained in Solution 1, , or . The area of the triangle is and the area of the hexagon is . Substituting in for , we get .
The area of an equilateral triangle with side length is .
The area of a regular hexagon with side length is .
Video Solution by OmegaLearn
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