If an equilateral triangle and a hexagon have the same perimeter, which area is greater and by how much? Please show work.
1 Answer
Let's say the perimeter of the equilateral triangle is
#A_"triangle" = (bh)/2#
From the diagram you can see that a 30-60-90 triangle has height
#A_"triangle" = (b*b/2*sqrt3)/2 = (b^2sqrt3)/4 = sqrt3 ~~ color(blue)(1.732)#
For the hexagon, you can think of it as six equilateral triangles of perimeter
With each of those triangles,
#A_"hexagontriangle" = (b^2sqrt3)/4 = sqrt3/4 ~~ 0.43#
And with six triangles in the hexagon, we get:
#= 6 * sqrt3/4 = (3sqrt3)/2 ~~ color(blue)(2.60)#
And so we get:
#((3sqrt3) / 2 - sqrt3)/(sqrt3) * 100% = color(highlight)"50% larger area"#