Homework Question for Geometry!

35 points!

Please show all work

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

more detail

step-by-step explanation: