Question

If an equilateral triangle and a hexagon have the same perimeter, which area is greater and by how much? Please show work.

Answer 1

Let's say the perimeter of the equilateral triangle is `2 + 2 + 2 = 6`, while the perimeter of the regular hexagon is `1 + 1 + 1 + 1 + 1 + 1 = 6`. Then:

`A_"triangle" = (bh)/2`

From the diagram you can see that a 30-60-90 triangle has height `b/2*sqrt3`, so we have, with `b = 2`:

`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 `3` (draw straight lines connecting corners across the hexagon).

With each of those triangles, `b = 1`, and we get:

`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"`