Mathematics Question and Solution
Sir Mike Ambrose is the author of the question.
Calculating Blue Area + Red Area Exactly.
Let x be the side length of the regular octagon.
a = ⅛*180(8-2)
a =135°
a is the single interior angle of the regular octagon.
b = 135-½(180-135)
b = 135-22.5
b =112.5°
c² = 2x²-2x²cos135
c = 1.84775906502x cm.
Calculating x.
0.5*x²sin135+0.5*0.5x*1.84775906502xsin112.5 = 16+24√(2)
0.78033008589x² = 49.941125497
x² = 64
x = 8 cm.
Again, x is the side length of the regular octagon.
Calculating Area Blue.
2d² = 8²
d² = 32
d = 4√(2) cm.
e = 2d+8
e = (8√(2)+8) cm.
Area Blue is;
(½((8√(2)+8)+8)*4√(2))+(½(8)*(8√(2)+8))-(½*4*4)-(16+24√(2))
= 4√(2)(4√(2)+8)+(32√(2)+32)-8-16-24√(2)
= 32+32√(2)+32√(2)+32-24-24√(2)
= 64√(2)+64-24-24√(2)
= 40+40√(2)
= 40(1+√(2)) cm²
Calculating Area Red.
f² = 2(8)²-2(8)² cos135
f = 14.7820725202 cm.
g² = 14.7820725202²+4²-2*4*14.7820725202cos112.5
g = 16.7261622014 cm.
(16.7261622014/sin112.5) = (4/sinh)
h = 12.7643896827°
j = 135-h-22.5
j = 99.7356103173°
k = 180-j
k = 80.2643896827°
l = 180-k-45
l = 54.7356103173°
(8/sin80.2643896827) = (m/sin54.7356103173)
m = 6.62741699798 cm.
Area red is;
0.5*6.62741699798*8cos45
= 18.7451660041 cm²
It implies;
Blue Area + Red Area is;
40(1+√(2))+18.7451660041
= 96.5685424949+18.7451660041
= 115.313708499 cm² in exact decimal.
