Question

The area of a regular hexagon is 1500 square centimeters. What is its perimeter? Please show working.

Answer 1

The perimeter is approximately `144.24cm`.

Explanation:

A regular hexagon consists of 6 congruent equilateral triangles, so its area can be calculated as:

`A=6*(a^2sqrt(3))/4=3*(a^2sqrt(3))/2`.

The area is given, so we can solve an equation:

`3*(a^2sqrt(3))/2=1500`

to find the length of the hexagon's side

`3*(a^2sqrt(3))/2=1500`

Multiplying by `2`

`3*(a^2*sqrt(3))=3000`

Dividing by `3`

`a^2*sqrt(3)=1000`

For further calculations I take approximate value of `sqrt(3)`

`sqrt(3)~~1.73`

So the equality becomes:

`1.73*a^2~~1000`

`a^2~~578.03`

`a~~24.04`

Now we can calculate the perimeter:

`P~~6*24.04`

`P~~144.24`

Answer 2

`"perimeter"=144.17"cm"`

Explanation:

The hexagon can be split into 6 equilateral triangle.

Each triangle has area of `frac{1500"cm"^2}{6}=250"cm"^2`

If the length of each triangle is `l`, then the perimeter of the hexagon is simply `6l`.

Looking at 1 triangle, the area is given by half x base x height.

The base is `l`. The height is found by cutting the triangle into half and applying Pythagoras theorem.

`h^2+(l/2)^2=l^2`

`h=sqrt(3)/2l`

`"Area"=1/2*l*h`

`=1/2*l*sqrt(3)/2l`

`=sqrt(3)/4l^2`

`=250"cm"^2`

`l=24.028"cm"`

`"perimeter"=6l=144.17"cm"`