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

Question

Aaliyah Anderson · Accepted Answer

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:

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#

Caroline Aucoin · Answer

#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#

Sadie Adams · Answer

undefined To find the area of a regular hexagon, you can use the formula:

$$ \text{Area} = \frac{3\sqrt{3}}{2} \times s^2 $$

Where $ s $ is the length of one side of the hexagon.

Given that the perimeter of the hexagon is 24 feet and it has 6 sides, each side's length is $ \frac{24}{6} = 4 $ feet.

Now, substitute $ s = 4 $ into the formula:

$$ \text{Area} = \frac{3\sqrt{3}}{2} \times 4^2 $$

$$ \text{Area} = \frac{3\sqrt{3}}{2} \times 16 $$

$$ \text{Area} = 24\sqrt{3} $$

So, the area of the hexagon is $ 24\sqrt{3} $ square feet.