Mathematics Question and Solution
Let the two congruent lengths be 1 unit.
tan40 = a/1
a = 0.83909963118 unit.
cos40 = 1/b
b = 1.30540728933 units.
Calculating r, radius of the red inscribed circle.
(r/tan(20)) + (r/tan(25)) = 1.30540728933
r = 0.26684617092 units.
Red area is:
π(0.26684617092)²
= 0.22370300775 square units.
Calculating Purple Area.
cos40 = c/2
c = 1.53208888624 units.
sin40 = 1/d
d = 1.55572382686
Where c and d is the height and base of the rectangle respectively.
tan50 = e/1
e = 1.19175359259 unit.
(1.19175359259/sin60) = (f/sin80)
f = 1.35521218262 units.
g = c-f
g = 0.17687670362 units.
(1.19175359259/sin80) = (h/sin60)
h = 1.04801052091 units.
i = f-h
i = 0.30720166171 units.
j = i+g
j = 0.48407836533
sin40 = k/2
k = 1.28557521937 units.
l = d-k
l = 0.27014860749 units.
(0.30720166171/sin100) = (m/sin20)
m = 0.10669001746 units.
Area Purple is;
0.5*0.48407836533*0.27014860749 - 0.5*0.10669001746*0.30720166171cos60
= 0.06538654815 - 0.01419214314
= 0.05119440501 square units.
Area Purple ÷ Area Red to 3 decimal places is;
0.0511944050 ÷ 22370300775
= 0.22884987343
≈ 0.229
