Mathematics Question and Solution
Let the side length of the regular octagon be 1 unit.
Notice!
x, radius of the green inscribed circle is 1 unit.
a = ⅛*180(8-2)
a = ⅛(1080)
a = 135°
a is the single interior angle of the regular octagon.
y² = 2-2cos135
y = √(√(2)+2) units.
y = 1.847759065 units.
y is the radius of the bigger circle ascribing the green circle.
Therefore, the fraction of the bigger circle shaded is;
Area green circle ÷ Area bigger circle
πx² ÷ πy²
= x²÷y²
= 1²÷√(√(2)+2)²
= 1÷(√(2)+2)
= (√(2)-2)/(-2)
= ½(2-√(2))
