Question

What is the area of a hexagon where all sides are 8 cm?

Answer 1

Area `=96sqrt(3)` `cm^2` or approximately `166.28` `cm^2`

Explanation:

A hexagon can be divided into `6` equilateral triangles. Each equilateral triangle can be further divided into `2` right triangles.

Using the Pythagorean theorem, we can solve for the height of the triangle:

`a^2+b^2=c^2`

where:
a = height
b = base
c = hypotenuse

Substitute your known values to find the height of the right triangle:

`a^2+b^2=c^2`
`a^2+(4)^2=(8)^2`
`a^2+16=64`
`a^2=64-16`
`a^2=48`
`a=sqrt(48)`
`a=4sqrt(3)`

Using the height of the triangle, we can substitute the value into the formula for area of a triangle to find the area of the equilateral triangle:

`Area_"triangle"=(base*height)/2`

`Area_"triangle"=((8)*(4sqrt(3)))/2`

`Area_"triangle"=(32sqrt(3))/2`

`Area_"triangle"=(2(16sqrt(3)))/(2(1))`

#Area_"triangle"=(color(red)cancelcolor(black)(2) (16sqrt(3)))/(color(red)cancelcolor(black)(2)(1))#

`Area_"triangle"=16sqrt(3)`

Now that we have found the area for `1` equilateral triangle out of the `6` equilateral triangles in a hexagon, we multiply the area of the triangle by `6` to get the area of the hexagon:

`Area_"hexagon"=6*(16sqrt(3))`
`Area_"hexagon"=96sqrt(3)`

`:.`, the area of the hexagon is `96sqrt(3)` `cm^2` or approximately `166.28` `cm^2`.