Question

What is the area of a hexagon whose perimeter is 24 feet?

Answer 1

See a solution process below:

Explanation:

Assuming this is a regular hexagon (all 6 sides have the same length) then the formula for the perimeter of a hexagon is:

Substituting 24 feet for `P` and solving for `a` gives:

`24"ft" = 6a`

`(24"ft")/color(red)(6) = (6a)/color(red)(6)`

`4"ft" = (color(red)(cancel(color(black)(6)))a)/cancel(color(red)(6))`

`4"ft" = a`

`a = 4"ft"`

Now we can use the value for `a` to find the area of the hexagon. The formula for the area of a hexagon is:

Substituting `4"ft"` for `a` and calculating `A` gives:

`A = (3sqrt(3))/2(4"ft")^2`

`A = (3sqrt(3))/2 16"ft"^2`

`A = 3sqrt(3) * 8"ft"^2`

`A = 24sqrt(3)"ft"^2`

or

`A ~= 41.569"ft"^2`

Answer 2

`24 sqrt3 = 41.57` square feet

Explanation:

We need to assume that it is a regular hexagon - meaning that all the six sides and angles are equal,

If the perimeter is `24` feet, then each side is `24/6 = 4` feet

A hexagon is the only polygon which is made up of equilateral triangles.

In this hexagon, the sides of the hexagon and therefore the sides of the triangles are all `4` feet and the angles are each `60°`

Using the trig Area formula, `A = 1/2ab sin C`, we can calculate the area of the hexagon as:

`A = 6 xx 1/2 xx4xx4xxsin60°`

`= 48 sin 60°`

`= 48 xx sqrt3/2`

`=24 sqrt3`

If you calculate it you will get `41.57 " feet"^2`