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

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#.