Answer:
the octagon has a width of 32
the simple formula for the length of one side of the octagon is x = 0.4142w, where w is the width of the octagon and x is the width of an edge.
source: http://mathcentral.uregina.ca/QQ/database/QQ.02.06/martin1.html
so the edge is 13.2544
so the polygons bottom has a total length of
32 + 32 + 13.2544 = 77.2544
More interesting is the length we need to cut the green shapes into a simple big and a simple smaller triangle, or rather two of each.
We get the bottom length of both bigger, outer green triangles by calculating
77.2544 - 32 = 45.2544
we subtract 32 to eliminate the length under the octagon.
note that this is the sum of the bottom lengths of both bigger triangles
devide by 2
22.6272
now the bottom length of the smaller, inner triangles is just
(77.2544 - 45.2544 - 13.2544)/2 = 18.7456
total bottom length - bigTriLengh - octaEdge / 2
Now to the height of the bigger inner and the smaller outer green triangles.
the smaller ones is
(32 - 13.2544)/2 = 9.3728
I hope this step is obvious
the height of the bigger outer triangles is this + an octaSide
9.3728 + 13.2544 = 22.6272
Now let's just multiply the width of the triangles by the height, skip the divideBy2 part so that we got both twins left and right at once, and then sum up the smaller with the bigger ones to finally get the green shaded area.
22.6272*22.6272+9.3728*9.3728=
599.83955968
I realize that much of the work was indeed to prove that the triangles I cut have indeed the same height as width.
Rounded to the nearest whole number
the green area is 600 square units.
Have a nice day
Brainliest would be appreciated
If there are questions left, feel free to ask them
