Question 1209347
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Draw a segment from the center to each vertex of the octagon. 
It will produce 8 pizza slices so to speak.

{{{
drawing(400,400,0,3,0.75,3.75,

line(1,1,2,1),line(2,1,2.70711,1.70711),line(2.70711,1.70711,2.70711,2.70711),line(2.70711,2.70711,2,3.41421),line(2,3.41421,1,3.41421),line(1,3.41421,0.29289,2.70711),line(0.29289,2.70711,0.29289,1.70711),line(0.29289,1.70711,1,1),

line(1.5,2.20711,1,1),line(1.5,2.20711,2,1),line(1.5,2.20711,2.70711,1.70711),line(1.5,2.20711,2.70711,2.70711),line(1.5,2.20711,2,3.41421),line(1.5,2.20711,1,3.41421),line(1.5,2.20711,0.29289,2.70711),line(1.5,2.20711,0.29289,1.70711)
)
}}}
Each pizza slice is an isosceles triangle with an apex angle of 360/8 = 45 degrees


Let's zoom in on one of the pizza slices. 
I'll draw a line to split this slice into two mirror halves.
This splits the 45 degree angle into 45/2 = 22.5 degrees.
{{{
drawing(400,400,0,3,0.75,3.75,
line(0.3,1,2.7,1),line(2.7,1,1.5,3.4),line(1.5,3.4,0.3,1),line(1.5,3.4,1.5,1),
locate(1.4,2,"h"),locate(1.9,0.95,"0.5"),locate(0.9,0.95,"0.5"),locate(1.55,2.6,22.5^o),locate(1.15,2.6,22.5^o),
line(1.5,1.2,1.7,1.2),line(1.7,1.2,1.7,1)
)
}}}
h = unknown height of this pizza slice
h = apothem of the octagon
The side length of the octagon has been split in half to 1/2 = 0.5


tan(angle) = opposite/adjacent
tan(22.5) = 0.5/h
h*tan(22.5) = 0.5
h = 0.5/tan(22.5)
h = 1.207106781187 approximately
Please make sure that your calculator is set to degrees mode.


The area of each of the original slices would be
area = 0.5*base*height
area = 0.5*1*1.207106781187
area = 0.603553390593
8 times of which leads to a total area of 8*0.603553390593 = <font color=red>4.828427125</font> which is approximate.


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Another approach would be to use this formula
Area = 0.25*s^2*n*cot(180/n)
where "cot" represents "cotangent". Recall that cot = 1/tan
s = side length
n = number of sides


If we use this formula then we'll have s = 1 and n = 8
Area = 0.25*s^2*n*cot(180/n)
Area = 0.25*1^2*8*cot(180/8)
Area = 2*cot(22.5)
Area = 2*1/tan(22.5)
Area = <font color=red>4.828427125</font>


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Edit: after greenestamps made his post I realize that tan(22.5) can be determined exactly in terms of radicals.


Use the identity {{{tan(theta/2) = ""+-sqrt( (1-cos(theta))/(1+cos(theta)))}}} to determine that {{{tan(45/2) = sqrt( (1-cos(45))/(1+cos(45)))}}}
Note that tan(22.5) is positive so we ignore the minus outside the root.
After simplification you should get {{{tan(22.5) = -1+sqrt(2)}}} which leads to {{{cot(22.5) = 1+sqrt(2)}}}


So, area = 2*cot(22.5) = 2*(1+sqrt(2)) = <font color=red>1+2*sqrt(2)</font>


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Exact Answer: <font color=red>2+2*sqrt(2) square units</font> 
Approximate Answer: <font color=red>4.828427125 square units</font>
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