Mathematics Question and Solution
Let a be the radius of the circ.
Calculating a.
b = ½(12+2)
b = 7 units.
c = ½(4+6)
c = 5 units.
d = b-2
d = 7-2
d = 5 units.
It implies;
a² = c²+d²
a = √(5²+5²)
a = √(50)
a = 5√(2) units.
Again, a is the radius of the circle.
Calculating blue area.
e² = 6²+12²
e = 6√(5) units.
sinf = (0.5*6√(5))/(5√(2))
f = 71.5650511771°
g = 2f
g = 143.130102354°
h² = 4²+2²
h = 2√(5) units.
sinj = (0.5*2√(5))/(5√(2))
j = 18.4349488229°
k = 2j
k = 36.8698976458°
Area blue is;
(½*6*12)+(143.130102354π(5√(2))²/360)-(½*(5√(2))²sin143.130102354°)+(½*2*4)+(36.8698976458π(5√(2))²/360)-(½*(5√(2))²sin36.8698976458)
= 36+62.4522886198-15+4+16.0875277198-15
= 88.5398163396 square units.
Calculating pink area.
l² = 4²+12²
l = 4√(10) units.
sinm = (0.5*4√(10))(5√(2))
m = 63.4349488229°
n = 2m
n = 126.869897646°
o² = 6²+2²
o = 2√(10) units.
sinp = (0.5*2√(5))/(5√(2))
p = 26.5650511771°
q = 2p
q = 53.1301023542°
Area pink is;
(½*4*12)+(126.869897646π(5√(2))²/360)-(½*(5√(2))²sin126.869897646)+(½*2*6)+(53.1301023542π(5√(2))²/360)-(½*(5√(2))²sin53.1301023542)
= 24+55.3574358898-20+6+23.1823804501-20
= 68.5398163399 square units.
Therefore;
Blue Area - Pink Area is;
88.5398163396-68.5398163399
= 20 square units.
Or
A very smart approach is;
Area Blue - Area Pink is;
((12*6)+(4*2))-((12*4)+(6*2)
= (72+8)-(48+12)
= 80-60
= 20 square units.
