Mathematics Question and Solution
Sir Mike Ambrose is the author of the question.
a = ⅛*180(8-2)
a = 135°
a is the single interior angle of the regular octagon.
2² = 2b²
b = √(2) units.
c = 2+b
c = (2+√(2)) units.
tand = (2+√(2))/2
d = atan(1+½√(2))°
tand = 2/e
1+½√(2) = 2/e
e = 4/(2+√(2)) units.
f = c-e
f = (2+√(2))-4/(2+√(2))
f = ((2+√(2))²-4)/(2+√(2)) units.
2/(2+√(2)) = g/((2+√(2))²-4)/(2+√(2))
2/(2+√(2)) = g(2+√(2))/((2+√(2))²-4)
g(2+√(2))² = 2(2+√(2))²-8
g = 2-(8/(2+√(2))²) units.
Shaded Area Exactly in square units is;
Area rectangle with length (2+√(2)) units and width 2 units - Area triangle with height 2 units and base (4/(2+√(2))) units - Area triangle with height (((2+√(2))²-4)/(2+√(2)) units and base (2-(8/(2+√(2))²)) units.
2*(2+√(2))-½(2*4/(2+√(2)))-½(((2+√(2))²-4)/(2+√(2))*(2-(8/(2+√(2))²))
= ½(4(2+√(2))-(2*4/(2+√(2)))-(((2+√(2))²-4)/(2+√(2))*(2-(8/(2+√(2))²)))
= ½((8+4√(2))-(8÷(2+√(2)))-(((2+√(2))²-4)÷(2+√(2))*2-(8÷(2+√(2))²)))
= 4.18376618407 in decimal.
= (27√(2)-34) Exactly.
