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What would the area of a regular octagon be if the only measurement known is the perimeter which in this case is 32 units.

 Apr 22, 2019
 #1
avatar+18460 
+3

Octagon has 8 sides.       32/8 = 4 units side length

 

Area of octgon = 2(1 + sqrt2) s^2        (s = side length)  

 Apr 22, 2019
 #3
avatar+209 
+2

I actually need to use trig to find the area. Could you help me find the area using trig and special right triangles. I hope you understand what I am trying to do.

OwenT154  Apr 22, 2019
 #5
avatar+18460 
+2

Go here:   (this site even has your question as an example)

https://byjus.com/octagon-formula/

ElectricPavlov  Apr 22, 2019
 #2
avatar+106 
+2

256 units 

 Apr 22, 2019
 #4
avatar+209 
+1

Explain.

OwenT154  Apr 22, 2019
 #6
avatar+101870 
+2

We can divide the octagon into  16 equal   right triangles

We will have one angle of 22.5°.....with an opposite side of  4/2  = 2

The other angle will =  67.5°

Using the Law of Sines....we can find the  side, s,  opposite this angle

 

s / sin 67.5  =  2/sin 22/5

 

s  =   2 sin 67.5 / sin 22.5

 

Note that  sin 22.5  = cos 67.5  ....so we have

 

s  =  2 sin 67.5 / cos 67.5

 

s  = 2 tan 67.5

 

s  = 2 tan (135 / 2)  =  2   [ csc 135  - cot 135 ]  =  2 [ √2 - -1 ]   = 2√2 + 2

 

So....the area of the octagon  =

 

16 * (1/2)  product of legs  =

 

8 (2)(2√2 + 2)  =     32√2  + 32    units^2

 

cool cool cool

 Apr 22, 2019
 #7
avatar+4296 
+2

Non-Trig Solution:

 

Draw an octagon...like the diagram as shown below. Since the perimeter of the octagon is 32 cm, each of its side length are  32 / 8 = 4 centimeters.

 

Now, draw a square around it...forming four 45-45-90 degree triangles. The side length opposite to the 45 degree angles is \(x\sqrt{2}=4, x=\frac{4}{\sqrt{2}}=2\sqrt{2}\).

 

Find the area of each triangle, and multiply that by 4, yielding...\(4*4=16.\)

 

The area of the square on the other hand is \((4\sqrt{2}+4)^2=48+32\sqrt{2}.\)

 

Finally, (the area of the square) - (area of the four triangles = (area of the octagon).

 

Thus, we have \(48+32\sqrt{2}-16=\boxed{32+32\sqrt{2}}.\)

 

 Apr 23, 2019

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