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

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Answer 1 · 2018-03-17T18:16:35

See a solution process below:

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

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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:

enter image source here

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 · 2018-03-17T18:43:29

$24 sqrt3 = 41.57$ square feet

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 $41.57 " feet"^2$