Question

Find the area of a 6-gon with side length 12? Round to a whole number.

Answer 1

374

Explanation:

Area of regular hexagon=`(3sqrt3)/2a^2` where `a` is side length

Answer 2

This is approximately `374.12 " units"^2` to 2 decimal places

Rounded this gives `374" units"^2`

Explanation:

Objective is to find the area of `1/2` the triangle then multiply that by 12 to obtain the total area.

Area of a triangle is `1/2xx"base"xx"hight"`

The angle marked in blue is `(360^o)/6 = 60^o`
Consider just `1/2` of the triangle:

The sum of angles in a triangle is `180^o`

Angle ABC is `90^o` so angle BCA is `180^o-90^o-30 = 60^o`

Length AB can be determined from `tan(60^0)=(AB)/(BC)`

`tan(60^o)=(AB)/6`

The height `AB=6tan(60)`

But `tan(60) = sqrt(3)" "` as an exact value.

So height `AB=6tan(60)=6sqrt(3)`

Thus area of `DeltaABC = a= 1/2xx"base"xx"height"`

`color(white)("dddddddddddddddddd")a= 1/2xx color(white)("d")6 color(white)("d")xx color(white)("d")6sqrt(3) color(white)("ddd")=18sqrt(3)`

We have 12 of these in the 6-gon so the total area is:

Area of the whole `A=12xx18sqrt(3) = 216sqrt(3)`

This is approximately `374.12 " units"^2` to 2 decimal places
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Note that `216sqrt(3) = 3/2sqrt(3)xx12^2`

Matching the `3/2sqrt(3)color(white)(.)a^2` given by Briana M

color(white)(.)