regular octagon

shahar

shahar

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There is a regular octagon that his side is 6 unit.
1622210980808.png
What are the size of the area of the grey area in the octagon?
I know for example that the size HG is 6. but what the area of the grey rectangle that is grey and his side is HG? How I can know the size of the two other sides (that are not the one side that parallel HG that equal to 6) in that rectangle? How I proof the proof that show that sides what they are?
 
shahar said:
There is a regular octagon that his side is 6 unit.
View attachment 27512
What are the size of the area of the grey area in the octagon?
I know for example that the size HG is 6. but what the area of the grey rectangle that is grey and his side is HG? How I can know the size of the two other sides (that are not the one side that parallel HG that equal to 6) in that rectangle? How I proof the proof that show that sides what they are?
Click to expand...
What do we know about the legs of those triangles?
 
It is regular, then all sides are equal. For example, [MATH]AB = DE = 6[/MATH].

The angles insides the triangle are both [MATH]45^o[/MATH].

The length [MATH]HC = BE = 6(1 + \sqrt{2})[/MATH], (you can prove it with a little mathematics)

[MATH]\frac{HC - 6}{2}[/MATH] gives you the other sides of triangles. After that you will have all the lengths, and you can find the required area easily.
 
Last edited:
shahar said:
There is a regular octagon that his side is 6 unit.
View attachment 27512
What are the size of the area of the grey area in the octagon?
I know for example that the size HG is 6. but what the area of the grey rectangle that is grey and his side is HG? How I can know the size of the two other sides (that are not the one side that parallel HG that equal to 6) in that rectangle? How I proof the proof that show that sides what they are?
Click to expand...
Since we are dealing with regular octagon (all sides are equal ). So:

HG = GF = FE = ED = DC = CB = BA = AH = HG = 6

What are the measures of the internal angles → mFGH = mGHA = m HAB = mABC = mBCD = mCDE = mDEF = mEFG = ?
 
I find it:
Every angle is 145 hence the angle in triangle is 45 by looking on the trapezoid. And from Pythagorean theorem 36 = a sqaure + b square = 2(a square)
a = sqrt(18) - sqrt(2) ............?? ..............................INCORRECT
a*a/2 + a*6 + a*a/2
Double it gives:
A square + a square + 12a = 2(18 -2sqrt(18)sqrt(2) + 4) + 12(6 - sqrt(2)). From here what to do....
 
Last edited by a moderator:
shahar said:
I find it:
Every angle is 145 hence the angle in triangle is 45 by looking on the trapezoid. And from Pythagorean theorem 36 = a sqaure + b square = 2(a square)
a = sqrt(18) - sqrt(2)..........?? ..............................INCORRECT
a*a/2 + a*6 + a*a/2
Double it gives:
A square + a square + 12a = 2(18 -2sqrt(18)sqrt(2) + 4) + 12(6 - sqrt(2)). From here what to do....
Click to expand...
36 = a sqaure + b square = 2(a square)

36 = 2 * a2

a2 = 18

a = √18 = 3√2
 
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