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The area of the shaded region can be found out subtracting the area of the octogon from the area of the circle.
The area of the octogon could be found adding the areas of the triangles formed within octogon, joining the center of the circle with each vertex of the octogon. We notice that the octogon is split into 6 triangles. The central angle within each triangle measures 360 degrees/6 = 60 degrees. Since the sides that are joining the center of the circle wich each vertex are equal, being the radius of the circle, then the triangles inside octogon are equilateral triangles.
We'll calculate the area of an equilateral triangle:
A = r*r*sin 60/2
A = `r^(2)` `sqrt(3)` /4
Now, we'll multiply this value by 6, since inside octogon we have 6 equilateral triangle.
A octogon = 6`r^(2)` `sqrt(3)` /4
A octogon = 3`r^(2)` `sqrt(3)` /2
All we need now is to calculate the area of the circle:
A circle = `r^(2)` `pi`
Now we can determine the area of the shaded region:
A region = A circle - A octogon
A region = `r^(2)` (`pi` - 3`sqrt(3)`/2 )
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