SOLUTION: An equilateral triangle and a regular hexagon have the same perimeter . what is the ratio of the area of the hexagon to the area of the triangle ?
Express your answer as a common
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Question 267025: An equilateral triangle and a regular hexagon have the same perimeter . what is the ratio of the area of the hexagon to the area of the triangle ?
Express your answer as a common fraction
Answer by CharlesG2(834) (Show Source): You can put this solution on YOUR website!
An equilateral triangle and a regular hexagon have the same perimeter . what is the ratio of the area of the hexagon to the area of the triangle ?
Express your answer as a common fraction
perimeter triangle = 3 * L length of one side of triangle
perimeter hexagon = 6 * 1/2 * L length of one side of triangle = 3 * L
each side of hexagon is L/2
area of triangle = 1/2 * base * height = 1/2 * L * height
height^2 + (L^2)/4 = L^2
height^2 = (3L^2)/4
height of triangle = (L*sqrt(3))/2
area of triangle = 1/2 * L * (L*sqrt(3))/2 = 1/4 * L^2 * sqrt(3)
area of hexagon = 6 * 1/2 * L/2 * (L*sqrt(3))/4
area of hexagon = 6 * 1/2 * 1/2 * 1/4 * L^2 * sqrt(3)
area of hexagon = 6 * 1/4 * 1/4 * L^2 * sqrt(3) (6/4 is 3/2)
area of hexagon = 3/2 * 1/4 * L^2 * sqrt(3)
area of hexagon = 3/8 * L^2 * sqrt(3)
ratio of the 2 areas = (3/2 * 1/4 * L^2 * sqrt(3))/(1/4 * L^2 * sqrt(3))
ratio of the 2 areas = 3/2
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