SOLUTION: We cut a regular octagon ABCDEFGH out of a piece of cardboard. If $AB = 1$, then what is the area of the octagon?

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Question 1209347: We cut a regular octagon ABCDEFGH out of a piece of cardboard. If $AB = 1$, then what is the area of the octagon?
Found 3 solutions by math_tutor2020, greenestamps, ikleyn:
Answer by math_tutor2020(3816)   (Show Source): You can put this solution on YOUR website!

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

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.

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 = 4.828427125 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 = 4.828427125

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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 to determine that
Note that tan(22.5) is positive so we ignore the minus outside the root.
After simplification you should get which leads to

So, area = 2*cot(22.5) = 2*(1+sqrt(2)) = 1+2*sqrt(2)

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Exact Answer: 2+2*sqrt(2) square units
Approximate Answer: 4.828427125 square units
Answer by greenestamps(13198)   (Show Source): You can put this solution on YOUR website!


Here is another way to solve this problem, this time getting an exact answer.

Add 45-45-90 right triangles to alternating sides of the regular octagon to form a square.

The hypotenuses of those triangles are edges of the regular octagon, so they each have edge length 1; and so their legs all have length

The side length of the square is then .

Put the four added triangles together with their right angles at a common point to see that the combined area of the four triangles is the area of a square with side length 1.

Then the area of the regular octagon is the area of a square with side length , minus the are of a square with side length 1:



ANSWER: <<== typo corrected thanks to note from tutor @ikleyn

(which to several decimal places is equal to the answer obtained by the other tutor, 4.828417...)

NOTE: This problem shows a formula that is familiar to many geometry students who participate in math contests: the area of a regular octagon with side length s is

<<== typo corrected thanks to note from tutor @ikleyn
Answer by ikleyn(52776)   (Show Source): You can put this solution on YOUR website!
.
We cut a regular octagon ABCDEFGH out of a piece of cardboard.
If AB = 1, then what is the area of the octagon?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


            Here is another way to solve the problem. 


Let R be the radius of the circumscribed circle around our octagon.


Let's find its radius via the side length s.


The octagon consists of 8 congruent isosceles triangles, having 
the common vertex in the center.


Each such a triangle is an isosceles triangle with the lateral sides of the length R
and the angle at the vertex of 45°.  Write the cosine rule equation for such a triangle 

     =  +  - ,

     = ,

     = .    (1)


In our case with s = 1, the last formula takes the form

     =  =  =  =  =  = .    (2)


Now the area of one such a triangle is

     =  =  =  =  = .    (3)


For the area of the entire octagon, we should take the quantity (3)  8 (eight) times to get

     =   square units.    ANSWER

Solved.



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