Mathematics Question and Solution
a = ⅙(180(6-2))
a = ⅙(180*4)
a = 120°
a is the single interior angle of the regular hexagon.
b² = 2²+4²-2*2*4cos120
b = √(20+7)
b = 2√(7) units.
(2√(7)/sin120) = (2/sinc)
c = 19.1066053509°
d = 120-2c
d = 120-2*19.1066053509
d = 81.7867892983°
sin30 = e/4
e = 2 units.
f = 2+e
f = 2+2
f = 4 units.
g² = 4²+4²-2*4*4cos120
g = √(32+16)
g = √(48)
g = 4√(3) units.
Let r be the radius of the inscribed circle.
Calculating r.
sin60 = r/(4-r)
r = 4sin60-rsin60
r+rsin60 = 4sin60
r(1+sin60) = 4sin60
r = 1.8564064606 units.
Again, r is the radius of the inscribed circle.
Therefore, green area is;
2(area triangle with height 4 units and base 4sin120 units) + Area rectangle with length 4√(3) units and base width 4 units - Area triangle with height 2√(7) units and base 2√(7)sin81.7867892983 units - Area circle with radius 1.8564064606 units.
= 2(0.5*4*4sin120)+(4√(3)*4)-(0.5*(2√(7))²sin81.7867892983-π(1.8564064606)²
= 13.8564064606+27.7128129211-13.8564064606-10.8266978078
= 16.8861151133 square units.
