Mathematics Question and Solution
Calculating Total Shaded Area, Area Grey, Purple and Blue.
Notice.
KLMN is a square.
Calculating area grey.
a = 6-4
a = 2 units.
sinb = 2/6
b = asin(⅓)
b = 19.4712206345°
c²+2² = 6²
c = 4√(2) units.
Area Grey is;
(6*2)-½(4√(2)*2)-(19.4712206345π*6²/360)
= 0.22608138033 square units.
Calculating area purple.
d = (90-b)
d = 70.5287793655°
e = ½(d)
e = 35.2643896828°
f = e+b
f = 35.2643896828+19.4712206345
f = 54.7356103173°
g = ½(180-f)
g = 62.6321948414°
h = 90-g
h = 27.3678051587°
(4/sin27.3678051587) = (2/sinj)
j = 13.28826561°
k = j+h
k = 40.6560707687°
Area purple is;
(6*6)-(40.6560707687π*4²÷360)-(35.2643896828π*6²÷360)-(0.5*6²sin54.7356103173)-(0.5*2*4sin40.6560707687)
= 36-5.6766583667-11.0786347561-14.6969384567-2.60606777301
= 1.94170064749 square units.
Calculating area blue.
l = 4√(2)-2
l = 3.65685424949 units.
m² = 6²+6²-2*6*6cos35.2643896828
m = 3.63486535834 units.
n = 90-k
n = 90-40.6560707687
n = 49.3439292313°
o² = 4²+4²-2*4*4cos49.3439292313
o = 3.33937985499 units.
3.33937985499² = 3.63486535834²+3.65685424949²-2*3.65685424949*3.63486535834cosp
26.5843456639cosp = 15.4333713594
p = acos(15.4333713594/26.5843456639)
p = 54.5112153403°
Area blue is;
(0.5*3.63486535834*3.65685424949
cos54.5112153403)+(49.3439292313π*4*4÷360)-(0.5*4*4sin49.3439292313)+(35.2643896828π*6*6÷360)-(0.5*6*6sin35.2643896828)
= 5.41143744109+6.88971224766-6.06907266803+11.0786347561-10.3923048454
= 6.91840693142 square units.
Therefore, total shaded area is;
0.22608138033+1.94170064749+6.91840693142
= 9.08618895924 square units.
≈ 9.1 square units to 1 decimal place.
