Mathematics Question and Solution
Let the inscribed purple regular heptagon side be 1 unit.
a = ⅐(180*5)
a = ⅐(900)°
b = 180-⅐(900))
b = ⅐(360)°
c = 180-2(⅐(360))
c = ⅐(540)°
(1/sin⅐(540)) = (d/sin⅐(360))
d = 0.8019377358 unit.
e = 2(0.8019377358)+1
e = 2.60387547161 units.
f = 90-½(⅐(540))
f = ⅐(360)°
cos⅐(360) = g/2.60387547161
g = 1.62348980186 units.
h = 2g
h = 3.24697960372 units.
Where h is the base of the ascribed quadrilateral.
cos(180/7) = i/0.8019377358
i = 0.72252093395 unit.
j = 2i
j = 1.4450418679 units.
k = ½(3.24697960372-1.4450418679)
k = 0.90096886791 unit.
l = 1.4450418679+k
l = 2.34601073581 units.
Area Orange is;
0.5*2.34601073581*2.60387547161sin(180/7) - 0.5*1.4450418679*1.8019377358sin(180/7)
= 1.32523709643-0.5648896129
= 0.76034748353 square units.
Calculating Shaded Area.
sin(360/7) = m/0.8019377358
m = 0.62698016883 unit.
tan18 = n/0.62698016883
n = 0.203718206 unit.
o =2n
o = 0.407436412 unit.
Where o is the side length of the regular inscribed pentagon.
p = 180-(540/7)
p = (720/7)°
q = 90-p
q = (90/7)°
r = 90-q
r = (540/7)°
tan(540/7) = 0.62698016883/s
s = 0.14310413211 unit.
t = s+0.5+0.8019377358
t = 1.44504186791 units.
Shaded Area is;
0.5(1.44504186791+0.8019377358)0.62698016883-0.5*0.407436412²*5(1/(2tan(180/5))
= 0.70440582565-0.28560686991
= 0.41879895574 square units.
Shaded Area ÷ Area Orange to 2 decimal places is;
0.41879895574÷0.76034748353
= 0.5507994237
≈ 0.55
