Question

What is the perimeter of a regular hexagon that has an area of `54sqrt3` units squared?

Answer 1

The perimeter of the regular hexagon is `36` unit.

Explanation:

The formula for the area of a regular hexagon is

`A = (3sqrt3 s^2)/2` where `s` is the length of a side of the

regular hexagon. `:. (3cancel(sqrt3) s^2)/2= 54 cancel(sqrt3)` or

`3 s^2=108 or s^2=108/3 or s^2=36 or s=6`

The perimeter of the regular hexagon is `P=6*s=6*6=36`

unit. [Ans]

Answer 2

Perimeter: `6` units

Explanation:

A hexagon can be decomposed into 6 equilateral triangles:

If we let `x` represent the length of each side of such equilateral triangle.

The area of a triangle with sides of length `x` is
`color(white)("XXX")A_triangle=sqrt(3)/4x^2`
`color(white)("XXXXXXXXXXXXXXXXXXXX")`(See below for derivation)

The area of the hexagon is `6A_triangle` which we are told is `54sqrt(3)` square units.

`6 * sqrt(3)/4x^2=54sqrt(3)`

`rarr sqrt(3)/4x^2=9sqrt(3)`

`rarr 1/4x^2=9`

`rarr x^2=4 * 9=2^2 * 3^2=6^2`

`rarr x =6color(white)("XXX")`Note since `x` is a geometric length `x>=0`

The perimeter of the hexagon is `6x`
`rarr` Perimeter of hexagon `= 36`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Finding the perimeter of an equilateral triangle with sides of length `x`:

Heron'[s formula for the area of a triangle tells us that if the semi-perimeter of a triangle is `s` and the triangle has sides of lengths, `x`, `x`, and `x`, then
`"Area"_triangle =sqrt(s(s-x)(s-x)(s-x))`

The semi-perimeter is `s=(x+x+x)/2=(3x)/2`
So `(x-s)=x/2`
and
`"Area"_triangle=sqrt((3x)/2 * (x/2) * (x/2) * (x/2))=sqrt(3)/4x^2`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Answer 3

`36`

Explanation:

Let's start from an equilateral triangle with side `2`...

Bisecting the triangle results in two right angled triangles, with sides `1`, `sqrt(3)` and `2` as we can deduce from Pythagoras:

`1^2 + (sqrt(3))^2 = 2^2`

The area of the equilateral triangle is the same as a rectangle with sides `1` and `sqrt(3)` (just rearrange the two right angled triangles for one way to see that), so `1 * sqrt(3) = sqrt(3)`.

Six such triangles can be assembled to form a regular hexagon with side `2` and area `6 sqrt(3)`.

In our example, the hexagon has area:

`54 sqrt(3) = color(blue)(3)^2 * (6 sqrt(3))`

So the length of each side is:

`color(blue)(3) * 2 = 6`

and the perimeter is:

`6 * 6 = 36`